Mike’s OEIS Authored Sequences
by Michael Thomas De Vlieger, updated 21 January 2024, St. Louis, Missouri. First published 19 August 2014.
Click here to jump to abstracts for sequences I’ve authored. Latest papers: see the index.
About
This page stands to be renovated; at hundreds of entries the abstract roster is too long for most people and now these need their own pages.
I’ve contributed the following sequences to the Online Encyclopedia of Integer Sequences. Thus far I have contributed 492 accepted sequences including 338 original, 70 in collaboration, and 84 in service of the OEIS as of 4 September 2023. I’ve written Mathematica/Wolfram Language code for 9998 sequences (as of 4 September 2023, about 5.32% of all Mathematica programs in OEIS), and have edited b-files, etc., for about 3600 sequences. My first contribution was 11 June 2014. My first rejected sequence was A279528 in December 2016 (I did voluntarily recycle 3 A-numbers). My first retraction due to error was A299989 on 26 February 2018. A347286 was recycled in favor of extant A089576. A349981, a collaboration with Richard Ollerton, was scrapped 6 February 2022 because it was a binary sequence sparse with 1s.
Most of my work is done on a PC but I have coded sequences using my daughter's Raspberry Pi and the free version of Mathematica that comes with the system! Yes, you can get into it too for the price of a Raspberry Pi!
Check out my fave sequences page, which lists sequences with V-numbers. See end of page for drafts and sequences in the making.
On 16 February 2018, I began doing some work for Dr. Neil Sloane. Therefter, sequences I author are not necessarily motivated by my own research, but pertain to others' research papers and articles that reference the OEIS. I have processed hundreds of citations, linking papers to the OEIS in the sequences and in citation aggregation reference pages.
In December 2021, Dr. Sloane asked me to handle approvals for certain reviewers.
Subject Index
Click below to read about sequences I authored. In the descriptions below, you can click the sequences listed to jump to the OEIS entry. Click here to view a menu of Mike’s math pages. This subject index only contains sequences written before 9 June 2022.
Triangle read by rows: T(n,k) = π(pk × p(n + 1)): A284061.
Nonprime 1 ≤ k ≤ n such that n and k are coprime: A285788.
Partial products of prime factors of n with repetition: A285904.
Anomalous cancellation: Numerator A292288, denominator A292289, and cancelled digit A292393 pertaining to the proper fraction with the smallest denominator that has a nontrivial anomalous cancellation in base b.
Prime divisor restricted lexically earliest sequences (PDRLES or pearl sequences): constitutively restrained versions of the EKG sequence: A353916: E2347, A353917: E37. constitutively restrained versions of the Yellowstone sequence: A352097: Y1379, A352098: Y1, Y2347, Y24, Y37.
Partitions: number of recursive self-conjugate partitions of n: A321223.
Numbers k that have recursively symmetrical partitions having Durfee square with side length n: A322457. Algorithm that generates A322457: A322156. Indices of records in A321223: A323034, records A323035. Numbers m that have recursively self-conjugate prime signatures A330781. n − s² for partition P of n in reverse lexicographic order: A331553. Row n contains distinct terms in row n of A331553: A331478. Irregular table where row n lists the distinct smallest primes p of prime partitions of n: A333238. Binary encoding of A333238: A333259. Minimal-product and maximal-product partitions of n, restricted to 3-smooth numbers: A347860, A348599.
Factor encoding: Positional notation akin to A058481 denoting prime divisors of n: A276379; notation reversed and decoded: A273258; first differences of prime divisors p of n (“π-code”): A287352. Numbers m that have recursively self-conjugate prime signatures A330781.
XOR-triangles: Irregular triangle S(n,k) of number of zero-triangles of side length k in T(n): A333624. Interpret row m = S(m,k) for m in A334556 as row m in A067255: A333625. Rotationally symmetrical XOR-triangles (RSTs) with central zero-triangles (CZTs): A334769; CZT frame widths j: A334796, CZT side lengths k: A334770. Smallest smallest n that produce an RST with CZT of side length k: A334771. a(n) = frame width j of CZTs in the RSTs of A334769(n): A334796. Smallest numbers m that generate a CZT of frame width n in an RST: A334836. Interpret row m = S(m,k) for m in A334769 as row m in A067255: A334896. RSTs that have singleton zeros in a hexagonal matrix: A334930. RSTs that have an impure grain size 2: A334931. RSTs that have an impure grain size 3: A334932. RSTs for n that set records for the side length k of zero-triangles (not necessarily central): A335077.
Reduced residue system (totatives): A289172 totatives with maximal number of prime divisors, A285905, A286424. “Quincunx” patterns: T(n,k) = [(2 | n ∧ 2 | k) ∨ (3 | n ∧ 3 | k)]: A349297, A349298, inverse of A054521: A349317, T(n,k) = [(2 | n ∧ 2 | k) ∨ (3 | n ∧ 3 | k) ∨ (5 | n ∧ 5 | k)]: A350014.
Totient ratio φ(n)/n: T(n,k) with squarefree m in row n = π(gpf(m)) order of increasing φ(m)/m as k increases: A307540. Binary encoding of A307540: A307544. T(n,2) in above triangle: pn#/p(n − 1) A306237. m in row n such that positive φ(m)/m − ½ is minimized: A325236, m in row n such that positive ½ − φ(m)/m is minimized: A325237.
Primorials:
pn#/p(n − 1) A306237, number of fully prime complementary totative pairs of pn#: A285905; number of partially prime complementary totative pairs of pn# A286424. Row n lists the largest power e of the prime divisors p_m of primorial p_n# such that p_m^e ≤ p_n#: A287010.
Primorials pn# and squarefree numbers m with ω(m) = n: listed: A287483; counted: A287484; T(n,k) counted by pk# | m: A287691, greatest differences between indices of prime factors among squarefree p_n# ≤ m ≤ (p_(n + 1)# − 1) such that m | p_n# and ω(m) = n: A287692. Less than 2 pn#: A288784. Row n lists indices of primorials that produce A025487(n): A304886, row n lists indices of primorials that produce A002182(n): A306737.
Tensor products of power ranges of prime divisors of n bounded by n: power ranges 1..p^e | n: A275055, power ranges p^e ≤ n: A275280. Primorial base notation for numbers in A025487: A307056. Product of primorial (A025487) divided by largest primorial that divides such: A307107. Number of highly composite numbers in the interval pk# ≤ m < p(k+1)#: A307113. T(n,m) = number of k ≤ pn# such that Ω(k) = m, where k is a term in A025487: A307133.
Fibonacci: A336683 binary compactification of residues k found in Fibonacci sequence mod n.
Neutral numbers (necessarily composite nondivisors in the cototient of n): A300859 where records occur in A045763 (neutral counting function), A300914 record transform of A045763 (neutral counting function), A294492 Numbers that set records for A045763(n)/n (proportion of numbers 1 ≤ m ≤ n that are neutral), A243822 semidivisor counting function, A243823 semitotative counting function, A272618 irregular triangle of semidivisors, A272619 irregular triangle of semitotatives, A292867 highly semitotative numbers, A292868 records in A243823, A293555 highly ‘semidivisible’ numbers, A293556 records in A243822, A294575 semitotative-dominant numbers, A294576 odd semitotative-dominant numbers, A295221 semitotative parity numbers, A295523 Nonprime numbers for which there are more semidivisors than semitotatives. Characteristic function: A304571. Characteristic function of semidivisors: A304570, characteristic function of semitotatives: A304572.
Regular numbers (numbers m | ne with integer e ≥ 0; normally we consider 1 ≤ m ≤ n): A244052 indices of records, A244053 records, A244974, A243103, multiplicities of A243103, A283866, A275280, A275881, richness of 1≤k≤n A279907, richness of regular 1≤r≤n A280269, maximum richness in range of n A280274, population of richnesses of regular numbers m of n: A294306. Simple underlying function of A280274, A280363. Specifically divisors A276379, A275055, rejected A279528. A293555, A293556. A301892: A010846(A002182(n)). A301893 numbers that set records for A010846(n)/τ(n). Characteristic function: A304569. Characteristic function of semidivisors: A304570.
Concatenation of multiplicities of prime divisors of highly composite numbers: A245500. “Highly regular numbers” A244052: A288784 necessary but insufficient condition, A288813 “turbulent candidates”, i.e. terms in A288784 but not in A060735.
Concerning semicoprimes/semitotatives: A243823 semitotative counting function, highly semitotative numbers (indices of records in A243823) A292867, records in A243823 A292868, A294575 semitotative-dominant numbers, A294576 odd semitotative-dominant numbers, A295221 semitotative parity numbers A299990 balance between semidivisors and divisors, A299991 numbers that have more semidivisors than divisors, A299992 numbers with more than 1 distinct prime divisor that have more divisors than semidivisors, A300155 numbers that have as many semitdivisors as divisors, A300156 numbers that set records for dominance of semidivisors over divisors, A300157 records in A299990. A300858 = A243823 − A243822, A300860 records in A300858. A316991 numbers semicoprime to 2 and 7, A316992 numbers semicoprime to 3 and 5.
Highly composite and superabundant numbers: A301892: A010846(A002182(n)). A301413 A002182(n)/A002110(A108602(n)), A301414 primitive values k such that k × P is highly composite for some primorial P. A301415 Number of primorials P such that A301414(k) × P is highly composite, A301416 primitive values k such that k × P is superior highly composite for some primorial P. A305025 ω(m) for superabundant m. A305056A004394(n)/A002110(A305025(n)), A340014 primitive values k such that k × P is highly composite for some primorial P.A304234 Superior highly composite numbers that are superabundant but not colossally abundant, A304235 Colossally abundant numbers that are highly composite but not superior highly composite. A338786 Numbers in A166981 that are neither superior highly composite nor colossally abundant. A306737 highly composite numbers as a product of primorials (A002110), A306802 position of HCNs in A025487. Primorial base notation for numbers in A025487: A307056. Number of highly composite numbers m in the interval pk# ≤ m < p(k+1)#: A307113; Number of superabundant numbers m in the interval pk# ≤ m < p(k+1)#: A307327.
Indices of products of a multiset of a contiguous range of the smallest primes (A055932 and A025487): position of HCNs in A025487: A306802. Indices of A025487(n) in A055932: A331119. Indices of A002110(n) in A055932: A331938. Indices of HCNs in A025487: A332034. Indices of superabundant numbers in A025487: A332035. Indices of numbers both colossally abundant and superior highly composite in A025487: A332241.
Factorial number system: unit fractions A294168. Primorial number system: unit fractions A299989 (retracted).
Multiple-base number system: Greedy algorithm producing a partition of n such that all elements are unique and in A003586: A276380, row length of A276380: A277070, (canonic length of n was proposed as A277044 but already appears as A237442), numbers n for which A277070 ≠ A237442: A277071, Irregular triangle T(n,k) of the number of partitions of length k such that all of the members of the partition are unique and in A003586 (numbers of all possible DBNS representations of n of length k): A277045.
The “Wichita” function: mappings of recursive function f(k) := k → k − k/p across primes p | k, defining a poset P(k): A333959 terms of A334184 that are not unimodal, A334144 largest rank level of A334184, A334238 least index of largest rank level m in A334144.
“Idaho” numbers: A347287 A347284 products of A347288,
Chains of contiguous integers m starting from n (n ≤ m < b) wherein each m introduces a novel prime divisor q. Numbers b that are barriers to the chains: A334468. Numbers n that break through the barriers b pertaining to (n − 1): A334469. Since b are numbers with relatively few prime divisors, all of which are relatively small, we examine the greatest prime divisor of b: A333518.
Collatz: A291213 iteration allowing 3x + 1 for evens.
Number of unique ways to write n as a product of: 4 divisors: A291833 records, A291927 positions; 5 divisors: A291833 records, A291928 positions.
Concerning anti-divisors: A241556, A241557, A242028, A242029.
Powers of some numbers in base 60: A250073, A254334, A254335, A254336. Hamming numbers in base 60: A250089.
Decimal-base arithmetic A256577, A286300. Full Reptend Primes: A261773.
Digits of base-b representation of prime unit fractions: 1/5 A262114, 1/7 A262115.
Auxiliary sequences: A064413/EKG:
lpf: A348470. A277272: sum of prime factors: A349543. A347113: pq = j → k prime: A349405, prime j in order of appearance: A349411. Zumkeller’s A095258; A350741: records in A095258, A350928 = 2 + ΣA095258, A350929 = A350928/A095258.
Variants: A347113: {k : j | k ∨ k | j}: A349681.
Collaboration with Michel Marcus: A242028.
Collaboration with Jean-Marc Falcoz: A257350 (a(n+1) = a(n) + smallest nonzero (and unused) integer embedded in a(n) as a (not necessarily contiguous) subchain.)
Collaboration with Robert G. Wilson v: A265723, A265733, A266141-A266149, ( Number of n-digit primes in which n − 1 of the digits are {1-9})
Collaboration with Robert G. Wilson v and Michel Lagneau: A268702-A268707 ( Largest n-digit prime having at least n − 1 digits equal to {1, 3, 4, 6, 7, 9})
Collaboration with Antti Karttunen: A283990, A293230, A293233, A293430, A334238.
Collaboration with Antti Karttunen, Robert G. Wilson v: A304572, A333786, A333959, A334111, A334230, A335860.
Collaboration with Jamie Morken: A283427, A285905, A286941-A286942, A287917-A287918, A335260-A335261.
Collaboration with Amiram Eldar: A303358, A303359.
Collaboration with Peter Kagey: A333959, A334144, A334230, A334238, A334786.
Collaboration with David Sycamore: A333238, A333337, A335285, A336015-A336016, A336093, A336893, A345147, A346175, A353125, A355269, A356294, A358535.
Collaboration with David Sycamore and Rémy Sigrist: A355149.
Collaboration with David Corneth: A336015-A335016.
Collaboration with Scott Shannon: A361593.
Collaboration with Neil Sloane: A300051, A306888, A306896-A306899, A306905-A306906, A307280, A308008, A324151. (A331156-9, A331448).
Produced for OEIS: A299798, A300125, A300127-A300130, A300405, A302691, A308677-9, A309114-A309116, A316991, A318232, A318233, A323670, A324151, A324152, (A331156-A331159, A331448), A331190, A335718-A335720, A335793, A335861, A336070-2, A339610, A341905, A342851, A344903, A346142, A347951, A348794-A348823
Rejections: former A279528.
See end of page for drafts and sequences in the making.
Sequence abstracts.
Note: V-series numbers are my own catalog numbers for sequences I often use; these are not affiliated with OEIS. The V-series is non-sequential.
A243822: V0051: (Semidivisor Counting Function ξd) Number of “semidivisors” of n, numbers m < n that do not divide n but divide ne for some integer e > 1.
11 June 2014
{0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 2, 0, 2, 1, 0, 0, 4, 0, 2,…}
I had long envisioned entering this sequence. The fact that it was missing from the OEIS even in 2011 seemed to indicate that no one else studied neutral numbers, those numbers m ≤ n that neither divide n nor are coprime to n. Though the neutral numbers are counted by OEIS A045763 and easily computed via ξ(n) = n − (τ(n) + φ(n) − 1), people hadn't seemed to determine these two subsets. In my 2011 proof that is more focused on the application of neutral numbers to number bases, I'd found that there could only be two types of neutral number, since there are only three configurations of composite numbers m ≤ n in terms of the divisibility of its primes with respect to another composite number n. This sequence counts the number of rich or non-divisor regular numbers m ≤ n of n. It relates to A045763: A0243822(n) = A045763(n) − A243823(n). It also relates to the regular number sequence A010846: A243822(n) = A010846(n) − A000005(n). Semidivisors are "too rich" in the multiplicity of at least one prime divisor p that also divides n (all of the prime divisors of semidivisors must divide n) therefore a semidivisor cannot divide n, but can divide a regular multiple of n.
List of semidivisors of n: A272618. Records: A293556, indices of records: A293555.
A243823: V0061: (Semitotative Counting Function ξt) Quantity of “semitotatives” [of n], numbers m < n that are products of at least one prime divisor p of n and one prime q coprime to n.
11 June 2014 {0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 3, 3, 4, 0, 3, 0, 5,…}
This is a sister sequence to A243822 above. It counts the number of m < n that are the product of one or more primes p that divide n and one or more primes q that are coprime to n. It relates to the number of m < n that are neutral to n A045763: A243823(n) = A045763(n) − A243822(n).
List of semitotatives of n: A272619. Records: A292868, indices of records: A292867.
A244052: V0032: Highly regular numbers a(n) defined as positions of records in A010846: a(1) = 1 and a(n) is the least number k > a(n − 1) such that A010846(k) > A010846(a(n − 1)).
18 June 2014 {1, 2, 4, 6, 10, 12, 18, 24, 30, 42, 60, 84, …}
On 13 June 2014, I produced a very large map of arithmetic relations of m ≤ n for n ≤ 2520. I'd been producing maps like this since 2011, but had only reached n ≤ 120; this was now possible after the November 2013 Mathematica mapping algorithms. The original maps seemed to confirm my thinking about superior highly composite numbers / colossally abundant numbers being especially convenient tools for human arithmetic. In the June 2014 map, what stood out most were prominent lines of regular m ≤ n. Superior highly composite numbers weren't so much accentuated as a different series. For instance, n = 360 didn't seem as accentuated as 330 or 390. Strange numbers like 1890 appeared to be preferred over highly composite numbers in the same range. Looking more closely at the map, I noticed that these were numbers with a large proportion of regular numbers. Examining the data for A045763, I could pick out "highly regular numbers", those that set records for the number of regular m ≤ n. I wrote a program that would find these records. This algorithm was efficient up to A244052(54) = 60,060. At that scale, 10,000 numbers would require about 8 hours to process.
I conjecture that a number n will set a record for the number of regular m ≤ n in three stages:
1. Primorials π(k)# set records since the small primes π(i) with i ≤ k produce the most regular products m ≤ n at the scale of π(k)#.
2.
"Turbulent zone": Products π(k − 1)# × π(k + i) < n π(k)#, with 1 ≤ i, even some products of π(k − 2)# × π(k)π(k + 1), etc., set records until these products exceed integer multiples of n π(k)#, with 2 ≤ n < π(k + 1). The total number of prime divisors of these numbers must not exceed k if a(n) < 2π(k)#, and must not exceed the number of prime divisors of nπ(k)# if a(n) < nπ(k)#.
3. "Integer-multiple zone": Once
nπ(k)# > nπ(k − 1)# × π(k + 1), then the only recordsetters are nπ(k)#. These integer multiples nπ(k)# set records till n = π(k + 1).
With this conjecture, on 16 July 2014, I added projected terms A244052 to n = 149. David Corneth extended calculation to primorial 23# and a bit beyond, filling in missing terms in the "turbulent" zone just larger than p_k#. I added projected terms to 29# at 174 on 9 February 2015, and to 31# at term 228.
This sequence and its sister A244053 are the subject of a major study of mine. Take a look at validated data including over 500 terms here. The necessary-but-not-sufficient condition has become A288784. "Turbulent" candidates: A288813.
A244053: V0033: Let m = A244052(n) = n-th highly regular number; a(n) = number of numbers r ≤ m, all of whose prime divisors p also divide m.
18 June 2014 {1, 2, 3, 5, 6, 8, 10, 11, 18, 19, 26, 28, …}
These are the number m of regular numbers k ≤ m, the data that correlates with A244052. The projection based on my above conjecture appears here. David Corneth extended calculation to term 149 on 10 February 2015.
A244974 (Regular Sum Function g0(n)) Sum of numbers m ≤ n whose set of prime divisors is a subset of the set of prime divisors of n.
8 July 2014 {1, 3, 4, 7, 6, 16, 8, 15, 13, 30, 12, 45, …}
This is the sum m of the regular numbers k ≤ n, listed at OEIS A162306 in a flattened triangle. The regular number counting function appears at A045763.
A245500 Concatenation of multiplicities of prime divisors of highly composite numbers A002182(n).
24 July 2014 {0, 1, 2, 11, 21, 31, 22, 41, 211, 311, 221, 411, 321, …}
I've been familiar with highly composite and superior highly composite numbers since 2006; in my first studies of these numbers, the famous Flammenkamp studies of these numbers cropped up early. I adopted Flammenkamp's prime multiplicity notation in my own studies (except I place multiplicity μk of prime π(k) in the “place” normally associated with 10(n − 1), opposite of Flammenkamp's left-justified convention). I was surprised that his notation was not found at the OEIS, so submitted this sequence. The "place" or "channel" for each prime π(k) is saturated when μk > 9, thus the sequence is valid until n = 220, when the highly composite number is divisible by 210.
A241419 Number of numbers m ≤ n that have one prime divisor p > sqrt(n) such that p divides m.
8 August 2014 {0, 1, 2, 1, 2, 3, 4, 4, 2, 3, 4, 4, 5, 6, 7, 7, 8, 8, 9, 10,…}
In an effort to begin helping the OEIS incorporate number sequences found in math journals as opposed to my own amateur meanderings, I indentified a sequence described in the then-current issue of Integers. This sequence is the result of my Mathematica script that was assembled to produce a sequence from a formula in the journal.
A241556 Number of prime anti-divisors m of n.
8 August 2014 {0, 0, 1, 1, 2, 0, 3, 2, 1, 2, 3, 1, 3, 1, 1, 2, 5, 2, 3, 2,…}
In late July 2014, a string of anti-divisor sequences were seeing revisions and new sequences from several other authors, especially Chai Wah Wu. I read about anti-divisors about a year before coming to OEIS and played with them, but considered them unimportant to the study of number bases, and abandoned them. Having re-read the definition at A066272, I began writing Mathematica code based on Harvey P. Dale's for several new or edited sequences, many of these transforms or other manipulations of algebra, etc. Somehow despite sequences like A242966, no one seemed to have studied plain-old prime anti-divisors, so I wrote this sequence.
A241557 Numbers k that do not have prime anti-divisors.
8 August 2014 {1, 2, 6, 30, 36, 54, 90, 96, 114, 120, 156, 174,…}
In the wake of A241556 I noticed that there were numbers that had only composite anti-divisors, that these were fairly common. Thus this sequence.
A242028 Numbers k such that the least common multiple of the anti-divisors of k is less than k.
11 August 2014 {3, 4, 6, 9, 36, 54, 96, 216, 576, 1296, 69984, …} (Suggested by Michel Marcus, corrected by Chai Wah Wu)
In correspondence with Michel Marcus of France, he had suggested playfully that the first twelve terms of this new sequence were "for sale". So I took him up on his challenge and wrote Wolfram language that produced it. Noticing that the numbers were all 3-smooth, I noticed a trend among the terms, and produced a Kronecker product of 2a and 3b with a and b between 1 and 8, testing the matrix with the script. This resulted in finding 69984 (25 · 37). On 20 August, Chai Wah Wu found terms 11 through 16 using a method similar to a Kronecker product as stated above with with a and b between 1 and 40. He also found several terms that are not necessarily contiguous with the first 16 of the sequence. It's a pleasure to work with other math enthusiasts and professionals (I am not a professional) and broaden what we can compute regarding these sequences. This was my first true collaboration. The people that contributed perhaps more to this sequence than my beginning it come from all over the world. Math is fun, I hope those of you reading this page can get inspired by this sort of work. It's ideal for the young undergraduate math, computer science, or STEM student.
A242029 Number of anti-divisors m ≤ n of n that are coprime to n.
11 August 2014 {0, 0, 1, 1, 2, 0, 3, 2, 1, 2, 3, 1, 4, 2, 1, 2, 5, 2, 3, 2, …}
In line with my preoccupation with "arithmetic functions", I wrote this sequence in the wake of realizing that nobody had studied plain-old anti-divisors of n that are coprime to n. Armed with my neutral numbers theorems, I noticed that while an anti-divisor m ≤ n by definition could not divide n, it was often neutral or coprime to n. I wrote a series of proofs by attempting to solve k(x + ½) = n for even and odd k and n. It seems obvious that antidivisors must be coprime to prime n. The proofs led to the following realizations. Odd antidivisors k must be coprime to n. Even antidivisors k > 2 must be neutral to composite n, meaning k neither divides n nor is coprime to n. The even antidivisors k of even composite n must be semidivisors of n, meaning k is the product of at least two primes p that also divide n; one of these primes p = 2. The even antidivisors k > 3 of odd composite n must be semitotatives of n, meaning k is the product of at least one prime p that divides n, and one prime q that is coprime to n. The prime q = 2. Odd n > 1 have the antidivisor 2, which is the only prime even antidivisor, by definition coprime to odd n. This set of proofs is my second major set, and the first (only) set that has nothing to do with number bases.
A243103: Product of "n-regular" numbers m ≤ n whose prime divisors p also divide n.
19 August 2014 {1, 2, 3, 8, 5, 144, 7, 64, 27, 3200, 11, 124416, …}
This sequence is similar to A244974; it multiplies the set of regular numbers m ≤ n.
A250073: Powers of 2 written in base 60, concatenating the decimal values of sexagesimal digits.
11 November 2014 {1, 2, 4, 8, 16, 32, 104, 208, 416, 832, 1704, 3408, 10816, 21632, 43304, 90608, 181216, 362432, 1124904, …}
This draft sequence intends to present the powers of 2 sexagesimally, in a manner akin to the way we display sexagesimal digits on digital clocks, eliminating the colon (:).
A250089: 5-smooth numbers written in base 60, concatenating the decimal values of sexagesimal digits.
11 November 2014 {1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30, 32, 36, 40, 45, 48, 50, 54, 100, 104, 112, 115, 120, 121, 130, …}
This draft sequence intends to present the Hamming numbers in a manner akin to the way we display sexagesimal digits on digital clocks, eliminating the colon (:). It presents the output of a famous early computer science problem in a manner analogous to the way such numbers would have been written by the ancient Mesopotamians in cuneiform. Donald Knuth’s 1972 ACM article on the subject is cited.
Three similar sequences akin to A250073 and some of Neil Sloane’s sequences representing powers of prime divisors in transdecimal bases:
A254334: Powers of 3 written in base 60, concatenating the decimal values of sexagesimal digits.
28 January 2015 {1, 3, 9, 27, 121, 403, 1209, 3627, 14921, 52803, 162409, 491227, …}
This draft sequence intends to present the powers of 3 sexagesimally, in a manner akin to the way we display sexagesimal digits on digital clocks, eliminating the colon (:).
A254335: Powers of 5 written in base 60, concatenating the decimal values of sexagesimal digits.
28 January 2015 {1, 5, 25, 205, 1025, 5205, 42025, 214205, 1483025, 9023205, 45124025, 346032205, …}
This draft sequence intends to present the powers of 5 sexagesimally, in a manner akin to the way we display sexagesimal digits on digital clocks, eliminating the colon (:).
A254336: Powers of 10 written in base 60, concatenating the decimal values of sexagesimal digits.
28 January 2015 {1, 10, 140, 1640, 24640, 274640, 4374640, 46174640, 742574640, 11709374640, 125136174640, 2083602574640, …}
This draft sequence intends to present the powers of 10 sexagesimally, in a manner akin to the way we display sexagesimal digits on digital clocks, eliminating the colon (:).
The preceding 3 sequences were written in November along with A250073 but never submitted, instead they awaited the acceptability of A250073 before loading the draft list. These sequences were studied 21-22 June 2007 in a sketchbook using argam numerals for digits in bases n > 10 that exceed 10.
On 19 February 2015, I extended a sequence proposed by Johan Westin, A255011, using SketchUp. Diagrams associated with this sequence appear here.
A256577: Sum_{k ≥ 0} (dk)^2*10^k, where Sum_{k ≥ 0} (dk)*10^k is the decimal expansion of n.
2 April 2015 {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 250, …}
Let k = the decimal power of a place m in a decimal number n. Raise each decimal place value m to a power k + 1, allowing carry to the next digit when necessary.
A261773: Number of full reptend primes p ≤ n in base n. See the Sequence Page.
31 August 2015 {0, 1, 0, 2, 0, 2, 2, 1, 1, 2, 2, 3, 1, 2, 0, 5, 2, 4, 3, …}
This sequence counts the number a(n) of primes p ≤ n in base n whose unit fraction expansion (1/p) in base n is has period (p − 1).
Full reptend primes are also called long period primes, long primes, or maximal period primes.
Even square n have a(n) = 0, odd square n have a(n) = 1, since 2 is a full reptend prime for all odd n.
Odd n have a(n) ≥ 1, since 2 is a full reptend prime in all odd n whose period is 1, i.e., the maximal period (p − 1).
Are 2 and 6 the only numbers other than even squares for which a(n) = 0? Are 3, 10 and 14 the only numbers other than odd squares for which a(n) = 1? (From Robert Israel).
Crossref: A001913.
A262114: Digits of the base-b expansion of 1/5.
11 September 2015 {0, 0, 1, 1, 0, 1, 2, 1, 0, 3, 1, 1, 1, 2, 5, 4, 1, 4, 6, 3, 1, 7, 2, 2, 2, 4, 9, 7, 2, 7, 10, 5, 2, 11, 3, 3, …}
This sequence lists digits of reptends of 1/5 for bases b that are coprime to 5 and the single digit after the decimal point for bases b divisible by 5. The values of the digits are converted to decimal. The number of terms associated with a particular value of b are cyclical: 4, 4, 2, 1, 1, repeat. This is because the values are associated with b (mod 5), starting with 2 (mod 5). The expansion of 1/5 either terminates after one digit when b ≡ 0 (mod 5) or is purely recurrent in all other cases of b (mod 5), since 5 is prime and must either divide or be coprime to b. The period for purely recurrent expansions of 1/5 must be a divisor of φ(5) = 4, i.e., one of {1, 2, 4}.
b ≡ 0 (mod 5): 1 (terminating)
b ≡ 1 (mod 5): 1 (purely recurrent)
b ≡ 2 (mod 5): 4 (purely recurrent)
b ≡ 3 (mod 5): 4 (purely recurrent)
b ≡ 4 (mod 5): 2 (purely recurrent)
The expansion of 1/5 has a full-length period 4 when base b is a primitive root of p = 5.
Crossref: A004526, A026741, A130845, A262115.
Digits of 1/5 for the following bases:
2 0, 0, 1, 1
3 0, 1, 2, 1
4 0, 3
5* 1
6 1
7 1, 2, 5, 4
8 1, 4, 6, 3
9 1, 7
10* 2
11 2
12 2, 4, 9, 7
13 2, 7, 10, 5
14 2, 11
15* 3
16 3
17 3, 6, 13, 10
18 3, 10, 14, 7
19 3, 15
20* 4
...
Asterisks above denote terminating expansion; all other entries are digits of purely recurrent reptends.
Each entry associated with base b with more than one term has a second term greater than the first except for b = 2, where the first two terms are 0, 0.
Entries for b ≡ 0 (mod 5) (i.e., integer multiples of 5) appear at 11, 23, 35, ..., every 12th term thereafter.
A262115: Digits of the base-b expansion of 1/7.
11 September 2015 {0, 0, 1, 0, 1, 0, 2, 1, 2, 0, 2, 1, 0, 3, 2, 4, 1, 2, 0, 5, 1, 1, 1, 2, 5, 1, 4, 2, 8, 5, 7, 1, 6, 3, 1, 8, 6, 10, 3, 5, …}
This sequence lists digits of reptends of 1/7 for bases b that are coprime to 7 and the single digit after the decimal point for bases b divisible by 7. The values of the digits are converted to decimal. he number of terms associated with a particular value of b are cyclical: 3, 5, 3, 5, 2, 1, 1, repeat. This is because the values are associated with b (mod 7), starting with 2 (mod 7). The expansion of 1/7 either terminates after one digit when b == 0 (mod 7) or is purely recurrent in all other cases of b (mod 7), since 7 is prime and must either divide or be coprime to b.
The period for purely recurrent expansions of 1/7 must be a divisor of φ(7) = 6, i.e., one of {1, 2, 3, 6}.
b ≡ 0 (mod 7): 1 (terminating)
b ≡ 1 (mod 7): 1 (purely recurrent)
b ≡ 2 (mod 7): 3 (purely recurrent)
b ≡ 3 (mod 7): 6 (purely recurrent)
b ≡ 4 (mod 7): 3 (purely recurrent)
b ≡ 5 (mod 7): 6 (purely recurrent)
b ≡ 6 (mod 7): 2 (purely recurrent)
The expansion of 1/7 has a full-length period 6 when base b is a primitive root of p = 7.
Digits of 1/7 for the following bases:
2 0, 0, 1
3 0, 1, 0, 2, 1, 2
4 0, 2, 1
5 0, 3, 2, 4, 1, 2
6 0, 5
7* 1
8 1
9 1, 2, 5
10 1, 4, 2, 8, 5, 7
11 1, 6, 3
12 1, 8, 6, 10, 3, 5
13 1, 11
14* 2
15 2
16 2, 4, 9
17 2, 7, 4, 14, 9, 12
18 2, 10, 5
19 2, 13, 10, 16, 5, 8
20 2, 17
21* 3
...
Asterisks above denote terminating expansion; all other entries are digits of purely recurrent reptends.
Each entry associated with base b with more than one term has a second term greater than the first except for b = 2, where the first two terms are 0, 0.
Entries for b ≡ 0 (mod 7) (i.e., integer multiples of 7) appear at 21, 43, 65, ..., every 22nd term thereafter.
Crossref: A004526, A026741, A130845, A262114.
A272618: V0050: Irregular array read by rows: n-th row contains (in ascending order) the nondivisors 1 ≤ k < n such that all the prime divisors p of k also divide n.
3 May 2016 {0, 0, 0, 0, 0, 4, 0, 0, 0, 4, 8, 0, 8, 9, 0, 4, 8, 9, 0, 0, 4, 8, 12, 16, 0, 8, 16, 9, 4, 8, 16, 0, 9, 16, 18, …}
The “semidivisors” (nondivisor regulars bounded by n) of n, i.e., the list of composite nondivisors k < n whose prime divisors p also divide n. Crossref: Union of A027750 and nonzero terms of a(n) = A162306, thus A000005(n) + A243822(n) = A010846(n). The union of nonzero terms of a(n) and A272619 = A133995, thus A243822(n) + A243823(n) = A045763(n).
A272619: V0060: Irregular array read by rows: n-th row contains (in ascending order) the numbers 1 ≤ k < n such that at least one prime divisor p of k also divides n and at least one prime divisor q of k is coprime to n.
3 May 2016 {0, 0, 0, 0, 0, 0, 0, 6, 6, 6, 0, 10, 0, 6, 10, 12, 6, 10, 12, 6, 10, 12, 14, 0, 10, 14, 15, 0, 6, 12, 14, 15, 18, …}
The “semitotatives” of n, i.e., the list of composite nondivisors k < n having at least one prime divisor p that also divides n and at least one prime divisor q that is coprime to n. Crossref: The union of nonzero terms of a(n) and A272619 = A133995, thus A243822(n) + A243823(n) = A045763(n).
A273258: Write the distinct prime divisors p of n in the (π(p) − 1)-th place, ignoring multiplicity. Decode the resulting number after first reversing the code, ignoring any leading zeros.
28 August 2016 {1, 2, 2, 2, 2, 6, 2, 2, 2, 10, 2, 6, 2, 14, 6, 2, 2, 6, 2, 10, …}
This sequence writes A276379(n) then reverses the digits and decodes the result. It simulates a reversal transcription error. In effect, it clears any leading zeros and interprets the largest prime divisor p_b of n as p = 2, with the next largest now the next smallest, etc., until the smallest prime divisor p_a becomes the largest. If π(p_b) − π(p_a) = m, then p_a = prime(m + 1).
A275055: V1020: Irregular triangle read by rows listing divisors d of n in order of appearance in a matrix of products that arranges the powers of prime divisors p of n along independent axes.
14 July 2016 {1, 1, 2, 1, 3, 1, 2, 4, 1, 5, 1, 2, 3, 6, 1, 7, 1, 2, 4, 8, 1, 3, 9, 1, 2, 5, 10, 1, 11, 1, 2, 4, 3, 6, 12, 1, 13, …}
The matrix of products that are divisors of n is arranged such that the powers of the prime divisors range across an axis, one axis per prime divisor. Thus a squarefree semiprime has a 2-dimensional matrix, a sphenic number has 3 dimensions, etc. Generally, the number of dimensions for the matrix of divisors = ω(n) = A001221(n). Because of this, τ(n) * (mod ω(n)) = 0 for n > 1. This follows from the formula for τ(n). Prime divisors p of n are considered in numerical order. Product matrix of tensors T = 1, p, p^2, …, p^e that include the powers 1 ≤ e of the prime divisor p that divide n. Crossrefs: Cf. A027750, A000005 (row length), A000203 (row sums), A056538. This sequence sets the stage for A275280.
Triangle begins:
1;
1, 2;
1, 3;
1, 2, 4;
1, 5;
1, 2, 3, 6;
1, 7;
1, 2, 4, 8;
1, 3, 9;
1, 2, 5, 10;
1, 11;
1, 2, 4, 3, 6, 12;
1, 13;
1, 2, 7, 14;
1, 3, 5, 15;
1 2, 4, 8, 16;
1, 17;
1, 2, 3, 6, 9, 18;
...
2 prime divisors: n = 72
1 2 4 8
3 6 12 24
9 18 36 72
thus a(72) = {1, 2, 4, 8, 3, 6, 12, 24, 9, 18, 36, 72}
3 prime divisors: n = 60
(the 3 dimensional levels correspond with powers of 5)
level 5^0: level 5^1:
1 2 4 | 5 10 20
3 6 12 | 15 30 60
thus a(60) = {1, 2, 4, 3, 6, 12, 5, 10, 20, 15, 30, 60}
4 prime divisors: n = 210
(the 3 dimensional levels correspond with powers of 5,
the 4 dimensional levels correspond with powers of 7)
level 5^0*7^0: level 5^1*7^0:
1 2 | 5 10
3 6 | 15 30
level 5^0*7^1: level 5^1*7^1:
7 14 | 35 70
21 42 | 105 210
thus a(210) = {1,2,3,6,5,10,15,30,7,14,21,42,35,70,105,210}
A275280: V3020: Irregular triangle listing numbers m of n that have prime divisors p that also divide n, in order of appearance in a matrix of products that arranges the powers of prime divisors p of n along independent axes.
28 July 2016 {1, 1, 2, 1, 3, 1, 2, 4, 1, 5, 1, 2, 4, 3, 6, 1, 7, 1, 2, 4, 8, 1, 3, 9, 1, 2, 4, 8, 5, 10, 1, 11, 1, 2, 4, 8, 3, 6, 12, 9, 1, 13, …}
Product matrix of tensors T = 1, p, p^2,..., p^e that include the powers 1 ≤ e of prime divisors p such that p^e ≤ n. This sequence is analogous to A275055 but differs from it in that the tensors T include not only powers p^e that divide n but all powers p^e ≤ n. The matrix a(n) is bounded by n, thus all products m ≤ n. Let ω(n) = A001221(n). The matrix a(n) has omega(n) dimensions and is an omega(n)-dimensional simplex with (ω(n) − 1) right-angle sides and 1 irregular surface that is bounded by n. A027750(n) is a subset of A162306(n) and in a(n), the terms of A275055(n) appear in an contiguous ω(n)-dimensional parallelepiped (parallelotope) with 1 at the origin and n at the opposite corner. Thus the ω(n)-dimensional array described by A275055(n) is fully contained in the simplex-like matrix described by a(n). Divisors appear within the parallelepiped while nondivisors appear in the field outside the parallelepiped (see examples at A275280). Terms within the parallelepiped appear in A027750(n) while those outside appear in A272618(n). For a(2^x + 2) there is a term m = (n − 2); m ≠ (n − 1) except for n = 2, since GCD(n, n − 1) = 1. a(p^e) = A027750(p^e) = A162306(p^e) = A275055(p^e) for e ≥ 1. Crossrefs: Cf. A162306, A010846 (row length), A243103 (row product), A027750 (divisors of n), A000005 (number of divisors of n), A272618 (nondivisors m ≤ n that have prime divisors p that also divide n), A243822 (number of such nondivisors of n), A275055 (Product of tensor of prime divisor powers that are also divisors). This sequence describes the structure of the “regular matrix” of n, the significance of differences among the distinct primes p of n regarding the regular function r(n), and part of a proof regarding necessary-but-not-sufficient conditions for terms in A244052.
Triangle begins:
1;
1, 2;
1, 3;
1, 2, 4;
1, 5;
1, 2, 4, 3, 6;
1, 7;
1, 2, 4, 8;
1, 3, 9;
1, 2, 4, 8, 5, 10;
1, 11;
1, 2, 4, 8, 3, 6, 12, 9;
1, 13;
1, 2, 4, 8, 7, 14;
1, 3, 9, 5, 15;
1 2, 4, 8, 16;
1, 17;
1, 2, 4, 8, 16, 3, 6, 12, 9, 18;
...
2 prime divisors: n = 96
1 2 4 8 16 32 64
3 6 12 24 48 96
9 18 36 72
27 54
81
thus a(96) = {1,2,4,8,16,32,64,3,6,12,24,48,96,9,18,36,72,27,54,81}.
The divisors of 72 (thus the terms of A275055(72)) appear in a rectangle delimited by 1 at top left and 72 at bottom right.
3 prime divisors: n = 60
(the 3 dimensional levels correspond with powers of 5)
level 5^0: level 5^1: level 5^2:
1 2 4 8 16 32 | 5 10 20 40 | 25 50
3 6 12 24 48 | 15 30 60 |
9 18 36 | 45 |
27 54 | |
thus a(60) = {1,2,4,8,16,32,3,6,12,24,48,9,18,36,27,54,5,10,20,40,15,30,60,45,25,50}.
The divisors of 60 (thus the terms of A275055(60)) appear in a parallelepiped delimited by 1 at top left of level 5^0 and 60 at bottom right of level 5^1
A275881: Numbers n such that A010846(n) ≥ n/2.
25 December 2016 {1, 2, 3, 4, 6, 8, 10, 12, 18, 30}.
Consider integers 1 ≤ r ≤ n such that all prime divisors p | r also divide n: call such numbers r “regular” to n. Divisors d | n are regulars r that themselves divide n along with their prime divisors, while “semidivisors” are nondivisor regular numbers r. This sequence is the finite set of positive integers n that are at parity or dominated by regular numbers. The number 3 has divisors {1, 3} and the nonregular 2. The number 8 has divisors {1, 2, 4, 8} and nonregular {3, 5, 6, 7}. The number 12 has divisors {1, 2, 3, 4, 6, 12} and semidivisors {8, 9} and nonregular {5, 7, 10, 11}. As n increases, the totient dominates the ranges of prime p > 3. The totient counts numbers (“totatives”) 1 ≤ t ≤ n coprime to n; with the exception of t = 1 (the empty product, a divisor and thus regular to all n), all totatives are nonregular since gcd(t, n) = 1. Though there are more regular r than divisors d for n > 30, the ratio of the regular counting function ratio r(n)/n = A010846(n)/n < ½ and generally diminishes as highly divisible n increases. The sequence A244052 shows numbers that set records for the regular counting function. The numbers that arrange the sequence are the primorials A002110. The regular counting function ratio r(n)/n decreases as the primorial increases. This is evident, looking at the way the regular counting function behaves, as seen in A275280 and the Mertens function method for counting regulars. The sequence is provably finite.
A276379: Write a “1” for each distinct prime divisor p of n in the (π(p) − 1)-th place, ignoring multiplicity.
2 September 2016 {0, 1, 10, 1, 100, 11, 1000, 1, 10, 101, 10000, 11, 100000, 1001, 110, 1, 1000000, 11, 10000000, 101, 1010, 10001, 100000000, …}
a(n) notes the distinct prime divisors p of n by writing “1” in the (π(n) − 1)-th place. Zeros hold the places of primes q less than the greatest prime divisor p that do not divide n. Thus a(n) consists of 1’s and 0’s like a binary number where each bit value, instead of representing 2^k, represents prime(k + 1). a(n) = A067255(n) with all nonzero digits converted to 1’s. a(n) = a(A007947(n)), that is, a number n shares a value of a(n) with the largest squarefree divisor A007947(n). Thus a(18) = a(6) = 11. a(p) = 1 in the leftmost place followed by (π(p) − 1) zeros. This function is akin to A067255(n) except we don’t note the multiplicity e of p in n, rather merely note “1” if e > 0. Unlike A067255(1024) = 10, there are no overflows in a(n) into the next place that encodes prime(p + 1) due to “carry”. 1024 = 2^10, thus a(1024) = a(2^e) = 1, with e ≥ 1 = 1. Crossrefs: A027748, A067255 (write multiplicity instead of 1 in the (π(p) − 1)th place), A079067 (reverse 0’s and 1’s in a(n) and convert to decimal), A087207 (a(n) interpreted as a binary number), A273258 (a(n) reversed and converted to decimal).
A276380: Irregular triangle where row n contains terms k of the partition of n produced by greedy algorithm such that all elements are in A003586.
25 September 2016 {1, 2, 3, 4, 1, 4, 6, 1, 6, 8, 9, 1, 9, 2, 9, 12, 1, 12, 2, 12, 3, 12, 16, 1, 16, 18, …}
The book “Multiple Base Number System” inspired this sequence. In that book, the authors devised a “two dimensional” number base b(2,3) that arranges binary on one axis and ternary on the other, with the empty product 1 as the origin. The places are thus two dimensional, with values that are generated by the Kronecker product of the prime power ranges ≥ 1 of 2 and 3 respectively. Therefore all the values in the matrix are regular to 6, i.e., are 3-smooth and in A003586. Because the only digits that appear in the system are 0 and 1, we can instead represent the two dimensional array as a sequence of the values in the array. This sequence does that. In effect we are looking for integer partitions of n such that no part is repeated and all parts are in A003586. Crossrefs: The 3-smooth numbers, i.e., numbers m whose prime divisors p divide 6: A003586. Length of “canonic” representation of n, i.e., the shortest partition of n such that all parts are unique and in A003586: A237442. Irregular triangle T(n,k) of the number of partitions of length k of n such that all parts are unique and in A003586: A277045. Row lengths of a(n): A277070. Numbers n such that a(n) ≠ A237442(n): A277071.
A277044: Recycled as it is already in the database as A237442.
Length of the most efficient or “canonic” DBNS representation of n. See A276380, A277045.
A277045: Irregular triangle T(n,k) read by rows giving the number of partitions of length k such that all of the members of the partition are distinct and in A003586.
27 September 2016 {1, 1, 1, 1, 1, 1, 0, 2, 1, 1, 1, 0, 2, 1, 1, 1, 1, 1, 2, 2, 0, 3, 1, 1, …}
This sequence represents all the integer partitions of n of length k such that no part is repeated and all are 3-smooth (i.e., regular to 6). The book “Multiple Base Number System” inspired this sequence (see A276380). Thus row n of this irregular triangle counts the number of solutions of length k that code for n in the DBNS. Note that A276380(n) does not necessarily generate the most efficient or “canonic” DBNS representation of n with n ≥ 41. The canonic representation has length A237442(n) and at times can involve 2 or more alternative renditions. See A276380 above for crossrefs.
A277070: Row length of A276380(n).
27 September 2016 {1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 2, 2, 2, 1, 2, 1, 2, …}
This sequence represents the partition size generated by greedy algorithm at A276380(n) such that all parts k are unique and in A003586. Compare this to A237442(n). See A276380 above for crossrefs.
A277071: Numbers n for which A277070(n) does not equal A237442(n).
27 September 2016 {41, 43, 59, 86, 88, 91, 113, 118, 123, 135, 155, 172, 176, 177, 182, …}
These are numbers n for which the greedy algorithm A276380(n) produces a partition of n with more than A237442(n) terms that are all unique and in A003586. A276380(n) = A237442(n) if n is in A003586. There may be more than one partition of n that has terms that are unique and in A003586. The first n in this sequence with that quality is n = 88. See A276380 above for crossrefs.
A279528: Numbers n such that tau(n) = n/2.
14 December 2016 {1, 2, 3, 4, 6, 8, 12}, REJECTED in favor of a note at A000005.
The finite set of positive integers that are at parity or dominated by divisors. The number 3 has divisors {1, 3} and the nondivisor 2. The number 8 has divisors {1, 2, 4, 8} and nondivisors {3, 5, 6, 7}. The number 12 has divisors {1, 2, 3, 4, 6, 12} and nondivisors {5, 7, 8, 9, 10, 11}. Dr. Sloane suggested that, because there are so many sequences that have these terms in their data fields, another sequence with them would only bring confusion, and I agree. This sequence would have helped set the stage for A275881.
A279907: V0035: Triangle read by rows: T(n,k) = smallest power of n that is divisible by k, or −1 if no such power exists.
26 December 2016 {0, 0, 1, 0, −1, 1, 0, 1, −1, 1, 0, −1, −1, −1, 1, 0, 1, 1, 2, −1, 1, 0, −1, −1, −1, −1, −1, 1, 0, 1, −1, 1, −1, −1, −1, 1, …}.
Consider integers 1 ≤ r ≤ n such that all prime divisors p | r also divide n: call such numbers r “regular” to n. Divisors d | n are regulars r that themselves divide n along with their prime divisors, while “semidivisors” are nondivisor regular numbers r. This sequence registers all nonregular numbers k as −1. For regular k = r, the sequence renders the least power ρ of n, r | nρ. This is called the “richness” ρ of n-regular r. The number r = 1 is the empty product; it divides n0 for all n. For divisors r = d, d | n¹ by definition, thus ρ = 1. Nondivisor regular r divide a power of nρ with ρ > 1. The idea for this sequence arose in considering the “PowerMod” method of determining n-regularity of 1≤k≤n. Originally I encountered Robert G. Wilson V’s algorithm in another sequence; this algorithm considered k regular to n if k | nk. Indeed this is true, however my observation is that n-regular 1≤r≤n divides a much smaller power nρ with ρ < k in all circumstances. This sequence shows that 1≤r≤n | ρ, with ρ relatively small.
The triangle T(n,k) begins (with −1 shown as "." for clarity):
n\k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 ...
1: 0
2: 0 1
3: 0 . 1
4: 0 1 . 1
5: 0 . . . 1
6: 0 1 1 2 . 1
7: 0 . . . . . 1
8: 0 1 . 1 . . . 1
9: 0 . 1 . . . . . 1
10: 0 1 . 2 1 . . 3 . 1
11: 0 . . . . . . . . . 1
12: 0 1 1 1 . 1 . 2 2 . . 1
13: 0 . . . . . . . . . . . 1
14: 0 1 . 2 . . 1 3 . . . . . 1
15: 0 . 1 . 1 . . . 2 . . . . . 1
...
A280269: V0036: Irregular triangle T(n,k) read by rows: smallest power ρ of n: nρ that is divisible by r, applied to terms in row n of A162306.
30 December 2016 {0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 2, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 2, 1, 3, 1, 0, 1, 0, 1, 1, 1, 1, 2, 2, 1, 0, 1, 0, 1, 2, 1, 3, 1, 0, 1, 1, 2, 1, 0, 1, 1, 1, 1, …}.
Consider integers 1 ≤ r ≤ n such that all prime divisors p | r also divide n: call such numbers r “regular” to n. Divisors d | n are regulars r that themselves divide n along with their prime divisors, while “semidivisors” are nondivisor regular numbers r. This sequence applies only to n-regular r shown in row n of A162306 and is equivalent to A279907 without its “mooted” negative terms. The sequence renders the least power ρ of n, r | nρ. This is called the “richness” ρ of n-regular r. The number r = 1 is the empty product; it divides n0 for all n. For divisors r = d, d | n¹ by definition, thus ρ = 1. Nondivisor regular r divide a power of nρ with ρ > 1. The idea for this sequence arose in considering the “PowerMod” method of determining n-regularity of 1≤k≤n. Originally I encountered Robert G. Wilson V’s algorithm in another sequence; this algorithm considered k regular to n if k | nk. Indeed this is true, however my observation is that n-regular 1≤r≤n divides a much smaller power nρ with ρ < k in all circumstances. This sequence shows that 1≤r≤n | ρ, with ρ relatively small.
Triangle T(n,m) begins: Triangle A162036(n,k):
1: 0 1
2: 0 1 1 2
3: 0 1 1 3
4: 0 1 1 1 2 4
5: 0 1 1 5
6: 0 1 1 2 1 1 2 3 4 6
7: 0 1 1 7
8: 0 1 1 1 1 2 4 8
9: 0 1 1 1 3 9
10: 0 1 2 1 3 1 1 2 4 5 8 10
...
A280274: V0037: a(n) = maximum value in row n of A279907 (Also, maximum value in row n of A280269).
30 December 2016 {0, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 3, 2, 1, 1, 4, 1, 2, 2, 4, 1, 2, 1, 4, 1, 2, 1, 4, 1, 1, 3, 5, 2, 3, 1, 5, 3, 2, …}.
Consider integers 1 ≤ r ≤ n such that all prime divisors p | r also divide n: call such numbers r “regular” to n. Divisors d | n are regulars r that themselves divide n along with their prime divisors, while “semidivisors” are nondivisor regular numbers r. This sequence applies only to n-regular 1≤r≤n. Consider the least power ρ of n such that r | nρ. This is called the “richness” ρ of n-regular r. Now regard the largest value of ρ for all n-regular 1≤r≤n. This number ρ is fairly easy to calculate without determining n-regular 1≤r≤n. Let p
be the least prime factor of n (i.e., lpf(n or A020639(n)). The largest possible value for ρ must be at most the largest possible exponent of the largest possible power of the smallest distinct prime divisor of n. This is because logp n is largest when p is smallest. Therefore, it is fairly easy to determine pe ≤ n. We can simply compute ρ thus. For r = 1, ρ must be 0 since 1 | n0. For r that is a prime or prime power, i.e., with ω(n) = 1, ρ = 1 since all regular r divide n. Further, ceiling( floor(logp n)/e ) = 1, with e being the multiplicity of lpf(n) in n. The largest possible value of ρ pertains to squarefree n since the maximum multiplicity in n is 1 by definition, and the numerator of ceiling( floor(logp n)/e ) is maximized. It is clear that a better “surefire” test for n-regular 1≤k≤n is not ne (mod k) with e = k, but e = ceiling( floor(logp n)/e ). This has fine-tuned the PowerMod test for regularity.
Row n of A280269 a(n)
1: 0 0
2: 0 1 1
3: 0 1 1
4: 0 1 1 1
5: 0 1 1
6: 0 1 1 2 1 2
7: 0 1 1
8: 0 1 1 1 1
9: 0 1 1 1
10: 0 1 2 1 3 1 3
11: 0 1 1
12: 0 1 1 1 1 2 2 1 2
13: 0 1 1
14: 0 1 2 1 3 1 3
15: 0 1 1 2 1 2
16: 0 1 1 1 1 1
...
A280363: V0230: a(n) = floor(logp n) with e = A020639(n), the least prime factor of n.
1 January 2016 {0, 1, 1, 2, 1, 2, 1, 3, 2, 3, 1, 3, 1, 3, 2, 4, 1, 4, 1, 4, 2, 4, 1, 4, 2, 4, 3, 4, 1, 4, 1, 5, 3, 5, 2, 5, 1, 5, 3, 5, …}.
This sequence underlies A280274 above. It gives the largest power e of the smallest prime divisor p of n that is still less than or equal to n itself. For n = 1, e = 0 since 1 is the empty product and no primes are less than or equal to 1. For n = prime p, e = 1 since the smallest prime divisor of p is p¹ itself, and the exponent is 1. For prime power pm, e = m by definition. For all other cases, e = floor(logp n). See also A280274.
A283866: Multiplicities of prime factors of A243103(n).
17 March 2017 {0, 1, 1, 3, 1, 4, 2, 1, 6, 3, 7, 2, 1, 9, 5, 1, 7, 2, 4, 2, 10, 1, 14, 7, …}.
An irregular number triangle T(n,k) with 1≤ k ≤ω(n) that lists the multiplicities of the product of all numbers 1≤ m ≤ n | nρ with ρ ≥ 0. Another way to think about a(n) is a count of the instances of prime divisors p | n among the factors of all numbers 1≤ m ≤ n. Row lengths are ω(n) = A001221(n), number of 1≤ m ≤ n | nρ with ρ ≥ 0 = A010846(n), list of 1≤ m ≤ n | nρ with ρ ≥ 0 = A162306(n).
A284061: Triangle read by rows: T(n,k) = π(pk × p(n + 1)).
19 March 2017 {3, 4, 6, 6, 8, 11, 8, 11, 16, 21, 9, 12, 18, 24, 34, 11, 15, 23, 30, 42, 47, …}.
A number triangle T(n,k) as in the name. This sequence unifies several extant sequences: T(n,1) = A020900(n + 1), T(n,2) = A020901(n + 1), T(n,3) = A020935(n + 1), T(n,4) = A020937(n + 1). This sequence has application in generating a list of squarefree numbers pn# ≤ m ≤ p(n + 1)# − 1 such that ω(m) = n.
Rows 1 ≤ n ≤ 12 of triangle T(n,k):
3
4 6
6 8 11
8 11 16 21
9 12 18 24 34
11 15 23 30 42 47
12 16 24 32 46 53 66
14 19 30 37 54 62 77 84
16 23 34 46 66 74 94 101 121
18 24 36 47 68 79 99 107 127 154
21 29 42 55 79 92 114 126 146 180 189
22 30 46 61 87 99 125 137 160 195 205 240
Values of m = q × p_n#/prime(k) < p_(n+1)# with q = prime(T(n,k)):
prime(k)
2 3 5 7 11 13
6 | 5
30 | 21 26
p_(n+1)# 210 | 195 190 186
2310 | 1995 2170 2226 2190
30030 | 26565 28490 28182 29370 29190
510510 | 465465 470470 498498 484770 494130 487410
All terms m of row n have ω(m) = A001221(m) = n.
A285769: (Product of distinct prime factors)^(Product of prime exponents).
25 April 2017 {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 36, 13, 14, 15, 16, 17, 36, 19, 100, 21, 22, 23, 216, 25, 26, 27, 196, 29, 30, 31, 32, 33, 34, 35, 1296, …}.
Multiplicative analog to A088865. a(1) = 1 since 1 is the empty product; 1¹ = 1. a(p) = p since ω(p) = A001221(p) = 1 thus p¹ = p. a(pm) = pm since ω(p) = 1 thus pm is maintained. For squarefree n with ω(n) > 1, a(n) = n. For n with ω(n) > 1 and at least one multiplicity m > 1, a(n) > n. In other words, let a(n) = km, where k is the product of the distinct prime factors of n and m is the product of the multiplicities of the distinct prime factors of n. a(n) > n for n in A126706 since there are 2 or more prime factors in k and m > 1.
A285788: Irregular triangle T(n,m): nonprime 1 ≤ k ≤ n such that n and k are coprime.
26 April 2017 {1, 1, 1, 1, 1, 4, 1, 1, 4, 6, 1, 1, 4, 8, 1, 9, 1, 4, 6, 8, 9, 10, …}.
Row n is a subset of A038566(n) such that the union of a(n) and A112484(n) = A038566(n). Row lengths are A048864(n) = A000010(n) − (A000720(n) − A001221(n)), i.e., φ(n) − (π(n) − ω(n)). 1 appears in every row since 1 is not prime and coprime to all n. 4 is the smallest composite and appears first in row 5 since 4 divides 4. Rows that contain the single term 1 are in A048597; the largest n = 30 such that the only term is 1. For prime p, row p contains 1 and all composites k < p, since 1 < m < p are coprime to p.
Triangle begins:
n\m 1 2 3 4 5 6 7
---+------------------------
1: 1
2: 1
3: 1
4: 1
5: 1 4
6: 1
7: 1 4 6
8: 1
9: 1 4 8
10: 1 9
11: 1 4 6 8 9 10
12: 1
13: 1 4 6 8 9 10 12
14: 1 9
15: 1 4 8 14
16: 1 9 15
...
A285904: Partial row products of table A027746, prime factors with repetition, reversed.
28 April 2017 {1, 2, 3, 2, 4, 5, 3, 6, 7, 2, 4, 8, 3, 9, 5, 10, 11, 3, 6, 12, 13, 7, 14, 5, 15, 2, 4, 8, 16, …}.
T(n,1) = A006530(n); T(n,A001222(n)) = n.
n | T(n,*) | A027746(n,*)
---+----------------+----------------
1 | 1 | 1
2 | 2 | 2
3 | 3 | 3
4 | 2, 4 | 2, 2
5 | 5 | 5
6 | 3, 6 | 2, 3
7 | 7 | 7
8 | 2, 4, 8 | 2, 2, 2
9 | 3, 9 | 3, 3
10 | 5, 10 | 2, 5
11 | 11 | 11
12 | 3, 6, 12 | 2, 2, 3
13 | 13 | 13
14 | 7, 14 | 2, 7
15 | 5, 15 | 3, 5
16 | 2, 4, 8, 16 | 2, 2, 2, 2
17 | 17 | 17
18 | 3, 9, 18 | 2, 3, 3
19 | 19 | 19
20 | 5, 10, 20 | 2, 2, 5
A285905: a(n) = A275768(A002110(n)).
3 May 2017 {0, 0, 5, 26, 124, 852, 7550, 86125, 1250924, 23748764, …}.
The number of ways to express primorial pn# as (prime(i) + prime(j))/2 when (prime(i) − prime(j))/2 also is prime. In other words, the number of complementary totative pairs of pn#.
A286300: Square root of smallest square formed from n by incorporating all the digits of n in a new decimal number.
5 May 2017 {1, 5, 6, 2, 5, 4, 24, 9, 3, 10, 11, 11, 19, 12, 34, 4, 42, 9, 13, 32, 11, 15, …}.
Square root of less restrictive version of A091873: a(n) ≤ A091873(n). First difference between a(n) and A091873(n) is for n=13. a(13) = sqrt(361) = 19, while A091873(13) = sqrt(1369) = 37.
If n is square then a(n) = sqrt(n).
Table of the first 20 terms of related sequences:
n A068165 A091873 a(n)^2 a(n)
1: 1 1 1 1
2: 25 5 25 5
3: 36 6 36 6
4: 4 2 4 2
5: 25 5 25 5
6: 16 4 16 4
7: 576 24 576 24
8: 81 9 81 9
9: 9 3 9 3
10: 100 10 100 10
11: 121 11 121 11
12: 121 11 121 11
13: 1369 37 361 19
14: 144 12 144 12
15: 1156 34 1156 34
16: 16 4 16 4
17: 1764 42 1764 42
18: 1089 33 81 9
19: 169 13 169 13
20: 2025 45 1024 32
...
A286424: Number of partitions of pn# into parts (q, k) both coprime to pn#, with q prime and k nonprime, where pn# = A002110(n).
8 May 2017 {0, 0, 1, 1, 4, 110, 1432, 23338, 397661, 8193828, 212858328, 5941706227, …}.
Number of totative pairs (q, k) such that prime q + k nonprime = pn# and both gcd(q, pn#) = 1 and gcd(k, pn#) = 1, with pn < q <= π(pn#), where π(pn#) = A000849(n) − n = A048862(n). Primes pn < q <= π(pn#) are greater than the greatest prime factor of pn# = pn, and are thus coprime to pn#. By the definition of primorial, we need not consider p ≤ pn, as these p are divisors of pn#, i.e., gcd(p, pn#) = p. Since the totatives of m can be paired such that a + b = m, we need only determine if (pn# − q) is not prime in order to count pairs (q, k). a(n) < floor(A005867(n)/2).
a(n) <= A048862(n). The totative pair (q,1) = (pn# − 1, 1) is counted by a(n) for n in A057704, with (pn# − 1) appearing in A057705.
A287010: Triangle T(n, m): floor(log(A002110(n))/log(prime(m))).
31 Aug 2017 {1, 2, 1, 4, 3, 2, 7, 4, 3, 2, 11, 7, 4, 3, 3, 14, 9, 6, 5, 4, 4, 18, 11, 8, 6, 5, 5, 4, 23, 14, 9, 8, 6, 6, 5, 5, …}. Row n lists the largest power e of the prime divisors pm of primorial pn# such that pme ≤ pn#.
Triangle begins:
1: 1
2: 2 1
3: 4 3 2
4: 7 4 3 2
5: 11 7 4 3 3
6: 14 9 6 5 4 4
7: 18 11 8 6 5 5 4
8: 23 14 9 8 6 6 5 5
9: 27 17 11 9 8 7 6 6 6
10: 32 20 14 11 9 8 7 7 7 6
11: 37 23 16 13 10 10 9 8 8 7 7
12: 42 26 18 15 12 11 10 10 9 8 8 8
...
A287352: V0321: Irregular triangle T(n,k) = A112798(n,1) followed by first differences of A112798(n).
23 May 2017 {0, 1, 2, 1, 0, 3, 1, 1, 4, 1, 0, 0, 2, 0, 1, 2, 5, 1, 0, 1, 6, 1, 3, 2, …}.
(Personally known as “π-code”, whereas A067255 is known as “multiplicity notation”).
This is a method of coding the indices of all the prime factors p of n with multiplicity, in order from least to greatest. Irregular triangle T(n,k) = first differences of prime divisors p of n. Row lengths = Ω(n) = A001222(n). Row sums = A061395(n). Row maxima = A286469(n). We can concatenate the rows 1 ≤ n ≤ 28 as none of the values of k in this range exceed 9: {0, 1, 2, 10, 3, 11, 4, 100, 20, 12, 5, 101, 6, 13, 21, 1000, 7, 110, 8, 102, 22, 14, 9, 1001, 30, 15, 200, 103}; a(29) = {10}, which would require a digit greater than 9. a(1) = 0 by convention. a(0) is not defined (i.e., null set). a(n) is defined for positive nonzero n. a(p) = A000720(p) for p prime. a(pe) = A000720(p) followed by (e − 1) zeros. a(product(pe)) is the concatenation of the a(pe) of the unitary prime power divisors pe of n, sorted by the prime p (i.e. the function a(n) mapped across the terms of row n of A141809). a(A002110(n)) = an array of n 1s. T(n,k) could be used to furnish A067255(n). We read data in row n of T(n,k). If T(n,1) = 0, then write 0. If T(n,1) > 0, then increment the k-th place from the right. For k > 1, increment the k-th place to the right of the last-incremented place. T(n,k) can be used to render n in decimal. If T(n,1) = 0, then write 1. If T(n,1) > 0, then multiply 1 by A000720(T(n,1)). For k > 1, multiply the previous product by π(x) = A000720(x) of the running total of T(n,k) for each k. Ignoring zeros in row n > 1 and decoding the remaining values of T(n,k) as immediately above yields the squarefree kernel of n = A007947(n). Leading zeros of a(n) are trimmed, but as in decimal notation numbers that include leading zeros symbolize the same n as without them. Zeros that precede nonzero values merely multiply implicit 1 by itself until we encounter nonzero values. Thus, {0,0,2} = 1 × 1 × π(2) = 3, as {2} = pi(2) = 3. Because of this no row n > 1 has 0 for k = 1 of T(n,k).
The triangle starts:
1: 0;
2: 1;
3: 2;
4: 1, 0;
5: 3;
6: 1, 1;
7: 4;
8: 1, 0, 0;
9: 2, 0;
10: 1, 2;
11: 5;
12: 1, 0, 1;
13: 6;
14: 1, 3;
15: 2, 1;
16: 1, 0, 0, 0;
17: 7;
18: 1, 1, 0;
19: 8;
20: 1, 0, 2;
...
A287483: Squarefree numbers A002110(n) ≤ k < A002110(n+1) − 1 such that A001221(k) = n.
25 May 2017 {1, 2, 3, 5, 6, 10, 14, 15, 21, 22, 26, 30, 42, 66, 70, 78, 102, 105, 110, 114, …}.
Primorial pn# is the smallest squarefree number with n prime factors.
a(n) is a list of squarefree numbers with n prime factors greater than and including pn# but less than p(n + 1)#.
a(1) includes the smallest primes less than 6.
a(2) lists the first squarefree semiprimes (A006881) less than 30,
a(3) gives the smallest terms of A033992 less than 210, etc.
The sequence begins with 1 as it is equal to A002110(0) and has 0 prime factors. The first primes less than 6 come next, followed by the first squarefree semiprimes (A006881) less than 30 and the smallest terms of A033992 less than 210, etc.
Triangle begins:
n Row n
0: 1;
1: 2, 3, 5;
2: 6, 10, 14, 15, 21, 22, 26;
3: 30, 42, 66, 70, 78, 102, 105, 110, 114, 130, ..., 195;
...
In each row n, the squarefree terms m must have ω(m) = n.
A287484: Number of squarefree A002110(n) ≤ k < A002110(n) such that A001221(k) = n.
25 May 2017 {1, 3, 7, 19, 58, 152, 422, 995, 2359, 6294, 14507, 36370, 88198, 187786, 386993, 840033, 1901930, 3851372, 8088478, 16388857, 30001902, …}.
Primorial pn# is the smallest squarefree number with n prime factors.
a(n) is a list of squarefree numbers with n prime factors greater than and including pn# but less than p(n + 1)#.
a(1) counts the first primes less than 6.
a(2) counts the first squarefree semiprimes (A006881) less than 30,
a(3) counts the smallest terms of A033992 less than 210, etc.
Row lengths of A287483.
A287691: Triangle T(n,k): number of squarefree numbers A002110(n) ≤ m ≤ (A002110(n + 1) − 1) such that A001221(m) = n and m is divisible by A002110(k).
26 May 2017 {1, 2, 1, 2, 4, 1, 3, 7, 8, 1, 5, 12, 23, 17, 1, 6, 16, 44, 56, 29, 1, 9, 24, 78, 130, 139, 41, …}.
T(n,n) = 1 since pn# is the only primorial divisible by pn#. Maxima for the first rows are {1, 2, 4, 8, 23, 56, 139, 351, 707, 1637, 3782, 8843, 18442, 38103, 77355, 177358, 387470, ...} at positions {1, 1, 2, 3, 3, 4, 5, 5, 5, 6, 7, 7, 7, 9, 10, 10, 10, ...}.
The triangle starts:
n | 0 1 2 3 4 5 6 7 8 9 10
-------------------------------------------------------------
0 | 1
1 | 2 1
2 | 2 4 1
3 | 3 7 8 1
4 | 5 12 23 17 1
5 | 6 16 44 56 29 1
6 | 9 24 78 130 139 41 1
7 | 9 30 107 214 351 224 59 1
8 | 11 39 154 332 707 650 389 76 1
9 | 17 64 261 598 1475 1637 1489 640 112 1
10 | 21 82 378 902 2496 3155 3782 2505 1041 144 1
...
There are A287484(2) = 7 squarefree numbers m between p2# = 6 and p3# − 1 = 29: {6, 10, 14, 15, 21, 22, 26}. Of these, {15, 21} are divisible by p0# = 1, {10, 14, 22, 26} are divisible by p1# = 2, and {6} is divisible by p2# = 6. Thus, T(2,k) = {2, 4, 1}. Note that the terms {15, 21}, {10, 14, 22, 26}, and {6} pertaining to the above example appear in row n of A287483 sorted as {6, 10, 14, 15, 21, 22, 26}.
A287692: Triangle read by rows: T(n,k): is the greatest difference between prime factors among squarefree numbers A002110(n) ≤ m ≤ (p(n + 1)# − 1) such that ω(m) = n and m is divisible by pk#.
15 June 2017 {3, 2, 5, 2, 3, 9, 2, 3, 5, 18, 2, 2, 4, 7, 30, 2, 2, 3, 5, 10, 42, 2, 2, 3, 4, 6, 13, 60, 2, 2, 3, 4, 5, 8, 17, 77, …}.
T(n,1) is the greatest index of the smallest prime divisor p of terms m in row n. T(n,n) = A120941(n). Consider the use of A287352 as a method for formulating squarefree numbers with n distinct prime factors. The values in row n serve as a limit beyond which we need not search further for terms pn# ≤ m ≤ (p(n + 1)# − 1). A287352 defines a squarefree number using a sequence of nonzero positive terms, beginning with the index of the smallest prime factor, then listing differences between indexes of subsequent prime factors in order of their magnitude. We can direct increment to the largest prime index as long as the number m < p(n + 1), then increment the index before it, etc. to produce the entire tree of factors that code numbers m.
Triangle begins:
n\k| 1 2 3 4 5 6 7 8 9 10 11 12
---------------------------------------------------------
1 | 3
2 | 2 5
3 | 2 3 9
4 | 2 3 5 18
5 | 2 2 4 7 30
6 | 2 2 3 5 10 42
7 | 2 2 3 4 6 13 60
8 | 2 2 3 4 5 8 17 77
9 | 2 2 3 3 4 6 10 22 113
10 | 2 2 2 3 4 5 8 12 25 145
11 | 2 2 2 3 4 5 6 9 15 32 179
12 | 2 2 2 3 4 4 6 7 11 19 36 229
...
For n = 2, there are A287484(2) = 7 squarefree numbers p2# ≤ m ≤ (p3# − 1) such that ω(m) = n. These are {6, 10, 14, 22, 26, 15, 21}. These numbers m have A287352(m) = {{1,1}, {1,2}, {1,3}, {1,4}, {1,5}, {2,1}, {2,2}} respectively; the largest values in both positions are {2,5}, thus row n = 2 of a(n) is {2,5}.
A288784: V3200: Irregular triangle read by rows: T(n,m) is the list of numbers k*A002110(n) ≤ k × t < (k + 1) × A002110(n) such that A001222(k × t) = n, with 1 ≤ k < prime(n + 1).
15 June 2017 {1, 2, 3, 4, 6, 10, 12, 18, 24, 30, 42, 60, 84, 90, 120, 150, 180, 210, …}.
See Terms of a(n) with 1 <= n <= 653 in A002110, A060735, and A244052.
A060735 and A002110 are subsets. This sequence is a necessary but insufficient condition for A244052. Terms that are in A060735 and A002110 are also in A244052. The first terms of this sequence that are not in A244052 are {3, 4290, 881790, 903210, 1009470, 17160990, 363993630, 380570190, 406816410, 434444010, ...}. Primorial pn# is the smallest squarefree number with n prime factors. Consider the list of squarefree numbers t with n prime factors greater than and including pn# but less than 2pn#. Extend the list to include products k × t of this list with 1 ≤ k < prime(n+1) such that k × t < (k+1) × pn#. This list contains squarefree numbers k × t with n distinct primes and presumes that the number (k+1) × pn# serves as a “limit” beyond which k × t > (k+1)pn# are not in the sequence. Charts: Relation of A288784 with A002110, A060735, and A244052; Tree Associated with Computation of terms of A288813 via Directed Iteration of A287352(A002110(n)).
A288813: V3201: Irregular triangle read by rows: T(m, k) is the list of squarefree numbers A002110(m) < t < 2 × A002110(m) such that A001221(t) = m. (Personally known as the “turbulent candidates” in A288784).
24 June 2017 {3, 10, 42, 330, 390, 2730, 3570, 3990, 4290, 39270, 43890, 46410, 51870, 53130, 570570, …}.
See Relations between A288813, A288784, A002110, and A244052, including prime decompositions of terms of a(n) and all code used to generate the tables.
A002110, A060735, and A244052 are subsets.
a(n) = terms t of row m of A288784 such that pm# < t < 2 × pm#. The only odd term is 3; the only other term not ending in 10, 30, 70, or 90 in decimal is 42. All terms t in row m have A001221(t) = m and at least one prime q coprime to t such that q < A006530(t). Consider "tier" m and primorial pm#, let “distension” i = π(A006530(T(m, k))) − m and let “depth” j = m − π(A053669(T(m, k))) + 1. Distension is the difference in the index of gpf(T(m, k)) and π(m), while depth is the difference between the index of the least prime totative of T(m, k) and pi(m) + 1. We can calculate the maximum distension i given m and j via imax = A020900(m − j + 1) − m − j + 1. This enables us to use permutations of 0 and 1 values in the notation A067255 and produce a(n) with some efficiency.
The most efficient method of generating a(n) is via f(x) = A287352(x), i.e., subtracting 1 from all values in row x of A287352. We use a pointer variable to direct increment on f(pm#) = a constant array of m 1's, until we have exhausted producing terms pm# < t < 2 × pm#. This enables the generation of T(m, k) for 1 ≤ m ≤ 100.
Triangle begins:
n a(n)
1: 3
2: 10
3: 42
4: 330 390
5: 2730 3570 3990 4290
6: 39270 43890 46410 51870 53130
7: 570570 690690 746130 870870 881790 903210 930930 1009470
...
A289171: V3210: Irregular triangle T(n, k) read by rows with 1 ≤ k ≤ n: T(n, 1) = A020900(n − k + 1) − (n − k + 1) and T(n, k) = max(0, T(n − 1, k − 1) − 1) otherwise.
21 July 2017 {1, 1, 1, 2, 3, 1, 3, 2, 4, 2, 1, 4, 3, 1, 5, 3, 2, 6, 4, 2, 1, 7, 5, 3, 1, 9, 6, 4, 2, 9, 8, 5, 3, 1, 9, 8, 7, 4, 2, 9, 8, 7, 6, 3, 1, …}.
This triangle is the function j in the chart at A288813.
Triangle begins:
n a(n)
1: 0
2: 1
3: 1
4: 2
5: 3 1
6: 3 2
7: 4 2 1
8: 4 3 1
9: 5 3 2
10: 6 4 2 1
11: 7 5 3 1
12: 9 6 4 2
13: 9 8 5 3 1
14: 9 8 7 4 2
15: 9 8 7 6 3 1
16: 11 8 7 6 5 2
17: 13 10 7 6 5 4 1
18: 12 12 9 6 5 4 3
19: 13 11 11 8 5 4 3 2
20: 14 12 10 10 7 4 3 2 1
...
A289172: Irregular triangle read by rows: row n lists terms m of A038566(n) such that A001221(m) = A051265(n), with a(1) = 1.
11 August 2017 {1, 1, 2, 3, 2, 3, 4, 5, 6, 3, 5, 7, 2, 4, 5, 7, 8, 3, 7, 9, 6, 10, 5, 7, 11, 6, 10, 12, 3, 5, 9, 11, 13, 14, 15, 6, 10, 12, …}.
Consider A051265(n), the largest value of ω(m) for 1 ≤ m ≤ n such that gcd(m, n) = 1 (i.e., m is in the reduced residue system or RRS of n, or m is a totative of n). Row n of this sequence consists of m in RRS(n) such that ω(m) = A051265(n).
Triangle begins:
n T(n,m) A051265(n)
1: 1 0
2: 1 0
3: 2 1
4: 3 1
5: 2 3 4 1
6: 5 1
7: 6 2
8: 3 5 7 1
9: 2 4 5 7 8 1
10: 3 7 9 1
11: 6 10 2
12: 5 7 11 1
13: 6 10 12 2
14: 3 5 9 11 13 1
15: 14 2
16: 15 2
17: 6 10 12 14 15 2
18: 5 7 11 13 17 1
19: 6 10 12 14 15 18 2
20: 3 7 9 11 13 17 19 1
A291213: Start from the singleton set S = {n}, and unless 1 is already a member of S, generate on each iteration a new set where each odd number k is replaced by 3k+1, and each even number k is replaced by 3k+1 and k/2. a(n) is the total size of the set from the singleton through after the first iteration which has produced 1 as a member, inclusive.
26 August 2017 {1, 3, 36, 6, 20, 72, 1168, 11, 216, 35, 576, 143, 111, 2422, 1657, 19, 336, 378, 6253, 66, 51, 1167, 820, 241, 24096, 180, 18805, …}.
See comments at A290100. A290100(n) is the size of the set at the last iteration, while this sequence is the sum of sizes of all generations including the last iteration.
A291833: Records transform of A252665.
3 September 2017 {1, 2, 3, 4, 5, 7, 9, 12, 16, 18, 21, 28, 30, 37, 43, 51, 53, 59, 66, 92, 103, 150, …}.
Connections with
A033833.
A291834: Positions of records of A252665.
3 September 2017 {1, 4, 8, 12, 16, 24, 36, 48, 72, 96, 120, 144, 216, 240, 288, 360, 432, 480, 576, 720, 1080, 1440, 2160, 2520, 2880, 3600, 4320, 5040, …}.
Distinct from A033833; first term not in A033833 is a(24) = 2520. There appear to be increasingly many terms a(n) not in A033833 as n increases. The terms 2520, 7560, 25200, 221760, 665280, 8648640, ... are not in A033833 but are in A002182. The term 3600 is the smallest that is in neither A033833 nor A002182, but in A007416. The term 831600 is the smallest that is in none of the three aforementioned sequences. Conjectures based on a(n) < 10^7:
1. Numbers in a(n) are products of the first several consecutive primes p.
2. Outside of a(1), the least prime factor of a(n) has multiplicity > 1. This implies no primes, primorials, or squarefree a(n) for n > 1.
3. The greatest prime factor of a(n) generally has multiplicity 1. Note, however, exceptions in a(n) for n = {1, 2, 3, 5, 7, 9, 12, 13, 15, 17, 19, 26, 29, 33, 73, ...}.
4. The multiplicities of prime factors p of m generally decrease or stay the same as p increases.
See “Records and first positions of records in A252665” for more information.
A291927: Records transform of A218320.
6 September 2017 {1, 2, 3, 4, 5, 7, 9, 11, 15, 16, 20, 25, 27, 33, 36, 46, 50, 53, 77, 86, 118, 145, 158, …}.
Connections with
A033833.
A291928: Positions of records in A218320.
6 September 2017 {1, 4, 8, 12, 16, 24, 36, 48, 72, 96, 120, 144, 216, 240, 288, 360, 480, 576, 720, 1080, 1440, 2160, 2520, 2880, 3360, 3600, 4320, 5040, …}.
A292288: Numerators of smallest denominator of a proper fraction that has a nontrivial anomalous cancellation in base b.
13 September 2017 {3, 4, 7, 6, 11, 8, 15, 13, 16, 12, 23, 14, 27, 22, 21, 18, 35, 20, 39, 29, 34, 24, 47, …}.
For prime base p, (p + 1)/(p^2 + p) simplifies to 1/p by cancelling digit k = 1 in the numerator and denominator. This fraction is written "11/110" in base p and simplifies to "1/10" = 1/p.
Smallest bases b for which n/d, simplified, has a numerator greater than 1 are 51, 77, 92, ...
A292289: Smallest denominator of a proper fraction that has a nontrivial anomalous cancellation in base b.
13 September 2017 {6, 12, 14, 30, 33, 56, 60, 39, 64, 132, 138, 182, 189, 110, 84, 306, 315, 380, 390, 174, …}.
For prime base p, (p + 1)/(p^2 + p) simplifies to 1/p by cancelling digit k = 1 in the numerator and denominator. This fraction is written "11/110" in base p and simplifies to "1/10" = 1/p.
Smallest bases b for which n/d, simplified, has a numerator greater than 1 are 51, 77, 92, ...
A292393: Base-n digit k involved in anomalous cancellation in the proper fraction A292288(n)/A292289(n).
15 September 2017 {1, 1, 3, 1, 5, 1, 7, 4, 6, 1, 11, 1, 13, 7, 5, 1, 17, 1, 19, 8, 12, 1, 23, 6, 15, 13, 9, …}.
For prime base p, (p + 1)/(p^2 + p) simplifies to 1/p by cancelling digit k = 1 in the numerator and denominator. This fraction is written "11/110" in base p and simplifies to "1/10" = 1/p.
A292867: V0062: Indices of records in A243823. (Highly semitotative numbers)
2 October 2017 (originally conceived here 30 January 2015) {1, 8, 14, 16, 20, 22, 26, 28, 32, 38, 40, 44, 46, 48, 50, 52, 54, 56, 58, 62, 64, …}.
Except for A292867(1) = 1, all terms are even. Some conjectures:
1. The only prime powers pe in this sequence are {8, 16, 32, 64}.
2. Squarefree terms m appear throughout. (There are 261 squarefree values among the first 1261 terms.)
3. Terms that set records for ω(m) are 1, followed by 2e, with 3 ≤ e ≤ 6, then 2e × 3 with 6 ≤ e ≤ 8, then 27 × A002110(k) with k ≥ 1.
4. Primorials A002110(n) for n ≥ 6 appear in this sequence. The first primorials in m are terms 6 through 8 of A002110 (i.e., 30030, 510510, 9699690) at n = 419, 774, 1258, respectively.
5. Outside of a(n) with 2 ≤ n ≤ 21 and n = {29, 30}, all terms of A244052 are also in this sequence. This observation applies to the smallest 104 terms of A244052.
6. For very large n, all terms are also in A244052. For small n, few terms of A244052 appear and are separated by many other numbers. Since numbers m in A244052 are products of k primes, many of which are the smallest primes, φ is minimized and A010846(m) becomes infinitesimal in comparison to m. Therefore A243823(m) is tantamount to the cototient of m. The size of n required to observe this agreement between this sequence and A244052 is unknown.
A292868: V0063: Records in A243823.
2 October 2017 (originally conceived here 30 January 2015) {0, 1, 3, 4, 5, 6, 8, 9, 11, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 31, 32, …}.
A293555: V0052: Indices of records in A243822 (Highly semidivisible numbers).
22 October 2017 (Originally 30 January 2015) {1, 6, 10, 18, 30, 42, …}.
This sequence is similar to A244052, except the function under transform is A010846(n) − A000005(n).
A293556: V0053: Records in A243822.
22 October 2017 (Originally 30 January 2015) {1, 5, 6, 10, 18, 19, …}.
This sequence is similar to A244053, except the function under transform is A010846(n) − A000005(n).
A294306: V0038: Irregular triangle T(n, m) = total number of each value k in row n of A280269.
30 October 2017 {1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 3, 1, 2, 1, 3, 1, 1, 1, …}.
The numbers i in A162306(n) divide nk with k ≥ 0; these k are listed in row n of A280269.
T(n, m) = total number of each value that k takes in row n of A280269, with 0 ≤ m ≤ A280274(n).
a(1) = 1 and T(n, 0) = 1 for all n, since 1 is the empty product and divides n0.
a(p) = 1, 1, since the only divisors of p are 1 and p; 1 | p0, and p | p1.
a(pe) = 1, e, since the only numbers in A162306(pe) are 1 and pk for 1 ≤ k ≤ e.
Row length of a(n) > 2 for n with ω(n) > 1.
Row length = A280274(n) + 1.
Total of row n = A010846(n).
Sum of terms m = {0, 1} in row n = A000005(n).
Terms in row n of A294306 start at 1, generally quickly rise to a maximum, then gradually decline to 1 at m = A280274(n).
Triangle begins:
1: 1
2: 1 1
3: 1 1
4: 1 2
5: 1 1
6: 1 3 1
7: 1 1
8: 1 3
9: 1 2
10: 1 3 1 1
11: 1 1
12: 1 5 2
13: 1 1
14: 1 3 1 1
15: 1 3 1
16: 1 4
17: 1 1
18: 1 5 2 1 1
19: 1 1
20: 1 5 2
...
A294492: V0044: Numbers m that set records for the ratio A045763(n)/n.
1 November 2017 {1, 6, 10, 14, 18, 22, 26, 30, 42, 60, 66, 78, 90, 102, 114, 126, 138, 150, 210, 330, 390, 420, 510, 570, 630, …}.
These numbers have an increasing proportion of nondivisors in the cototient (A051953(n)) with respect to n.
In other words, these numbers have an increasing proportion of smaller numbers that are counted neither by τ or φ.
Conjectures:
1. Let k = any product of primorial A002110(i − 1) and the smallest i primes. All terms m are in A002110 or of the form k × p, with prime p ≥ prime(i) such that k < A002110(i + 1).
2. For m ≥ A002110(5) = 2310, all terms m are in A002110 or of the form prime p × A002110(n), with prime(1) ≤ p ≤ prime(i).
Table of terms less than A002110(6):
b(n) = A045763(n), c(n) = exponents of the smallest primes such that the product = n, e.g., "2 1 0 1" = 2^2 * 3^1 * 5^0 * 7^1 = 126.
n a(n) b(n) c(n)
1 1 0 0
2 6 1 1 1
3 10 3 1 0 1
4 14 5 1 0 0 1
5 18 7 1 2
6 22 9 1 0 0 0 1
7 26 11 1 0 0 0 0 1
8 30 15 1 1 1
9 42 23 1 1 0 1
10 60 33 2 1 1
11 66 39 1 1 0 0 1
12 78 47 1 1 0 0 0 1
13 90 55 1 2 1
14 102 63 1 1 0 0 0 0 1
15 114 71 1 1 0 0 0 0 0 1
16 126 79 1 2 0 1
17 138 87 1 1 0 0 0 0 0 0 1
18 150 99 1 1 2
19 210 147 1 1 1 1
20 330 235 1 1 1 0 1
21 390 279 1 1 1 0 0 1
22 420 301 2 1 1 1
23 510 367 1 1 1 0 0 0 1
24 570 411 1 1 1 0 0 0 0 1
25 630 463 1 2 1 1
26 1050 787 1 1 2 1
27 1470 1111 1 1 1 2
28 2310 1799 1 1 1 1 1
29 4620 3613 2 1 1 1 1
30 6930 5443 1 2 1 1 1
31 11550 9103 1 1 2 1 1
32 16170 12763 1 1 1 2 1
33 25410 20083 1 1 1 1 2
...
A294575: V0064: Numbers n such that 2 × A243823(n) > n. (“Semitotative-Dominant Numbers”; Numbers a(n) that have more semitotatives than a(n)/2).
17 November 2017 (written 30 January 2015) {144, 162, 174, 186, 192, 198, 200, 204, 216, 220, 222, 228, 230, 234, 238, 240, 246, 250, 252, 258, …, 945, …}
Semitotatives are composite numbers m in the cototient of n that are products of at least one prime divisor p of n and at least one prime that is coprime to n. Such a number m is said to be "semicoprime" to n.
This sequence lists numbers that have more semicoprime m < n than totative and regular m <= n combined. This list is a subset of the sequence directly above. The smallest odd term is 945, the smallest primorial 2310, superior highly composite number 360. This sequence does not contain primes, prime powers, and semiprimes.
144 is in the sequence, since there are 74 semitotatives {10, 14, 15, 20, 21, 22, 26, ..., 140, 141, 142}.
Crossrefs: A243823, A005101.
A294576: V6401: “Odd Semitotative-Dominant Numbers”; Odd m that have more semitotatives than m/2, i.e., 2 × A243823(m) > m.
30 January 2015 {945, 1155, 1365, 1575, 1785, 1995, 2145, 2205, 2415, 2625, 2805, 2835, 3003, 3045, 3135, 3255, 3315, 3465, 3675, 3705, …}
The smallest even term is 144. This monotonic sequence contains only the odd terms of the sequence directly above.
945 is in the sequence, since there are 477 semitotatives {6, 10, 12, 14, 18, 20, ..., 939, 940, 942}.
Crossrefs: A243823, A005101.
A295221: V6400: Numbers k such that 2 × A243823(k) = k. (“Semitotative-Parity Numbers”).
17 November 2017 {156, 190, 224, 286, 352, 416, 544, 578, 608, 736, 928, 992, 1184, 1312, 1376, 1504, 1696, 1888, 1952, 33555776, 33557824, 33558208, …}
Observations:
1. There is a large gap between a(19) and a(20).
2. Products 25 × prime(i), with 3 ≤ i ≤ 17, are in the sequence.
3. Products 26 × prime(j), with 43391 ≤ j ≤ 82025, are in the sequence.
4. a(1) = 2² × 3 × 13, and terms 190, 286, and 578 are even, but do not follow the pattern of 2h × p prime.
945 is in the sequence, since there are 477 semitotatives {6, 10, 12, 14, 18, 20, ..., 939, 940, 942}.
A295523: V0045: Nonprime numbers n such that A243822(n) ≥ A243823(n). (“Semidivisor-Favorable Numbers”).
23 November 2017 {1, 4, 6, 10, 12, 18, 30}
The sequence is finite with only these 7 terms.
Consider numbers m that are nondivisors in the cototient of n, listed in row n of A133995 and counted by A045763(n). This sequence lists numbers n for which there are more m such that m | ne with e ≥ 0 than there are m that are products of at least one prime divisor p of n and one nondivisor prime q. The former species of m are "semidivisors" listed in row n of A272618 and counted by A243822(n), while the latter are "semitotatives" listed in row n of A272619 and counted by A243823(n). These two species constitute the only species of nondivisors in the cototient of n.
Primes p have no nondivisors in the cototient, i.e., A045763(p) = 0, therefore A243822(p) and A243823(p) also are 0. The equality of these latter two sequences is trivial in the case of primes.
Prime powers pe except for pe = 4 have A243823(pe) > A243822(pe), since A243822(pe) = 0. All powers pk with 0 ≤ k ≤ e divide pe.
The sequence is finite because there exist a lot more nondivisor primes q than p | n as n increases. Therefore there are more numbers m in row n of A272619 than there are in row n of A272618, since the former are products p × q and the latter are products only of p.
1 is in the sequence because it is not prime and there are no nondivisors in the cototient, therefore A243822(1) = A243823(1) = 0.
4 is in the sequence because it is the very smallest composite; nondivisors in the cototient of n are composite and since 4 | 4, both A243822(4) and A243823(4) = 0.
6 is in the sequence because it is the only number for which A243822(6) = 1 but A243823(6) = 0. A272618(6) = 4; 4 | 6².
10 is in the sequence because it has 2 semidivisors 4 | 10² and 8 | 10³, while only 1 semitotative 6 = 2 × 3.
14 is not in the sequence since it has 2 semidivisors (4 and 8) but 3 semitotatives (6, 10, and 12).
List of terms n followed by row n of A272618 and A272619:
1, {}, {}
4, {}, {}
6, {4}, {}
10, {4,8}, {6}
12, {8,9}, {10}
18, {4,8,12,16}, {10,14,15}
30, {4,8,9,12,16,18,20,24,25,27}, {14,21,22,26,28}
A294168: Irregular triangle read by rows in which row n contains significant digits after the radix point for unit fractions 1/n expanded in factorial base.
10 February 2018 {0, 1, 0, 2, 0, 1, 2, 0, 1, 0, 4, 0, 1, 0, 0, 3, 2, 0, 6, 0, 0, 3, 0, 0, 2, 3, 2, 0, 0, 2, 2, 0, 0, 2, 0, 5, 3, 1, 4, …}
See the Wikipedia link for the construction method of 1/n in factorial base. This version eliminates the 1/0! and 1/1! places, which are always 0.
By convention, row n = 1 contains {0}.
Length of row n = A002034(n) − 1.
Length of row p = p − 1 for p prime.
1/n expanded in factorial base appears below; this sequence includes numbers to the right of the radix point.
n 1/n in factorial base A276350(n) A299020(n)
-- ---------------------- ---------- ----------
1 1.0 1 1
2 0.1 1 1
3 0.0 2 2 2
4 0.0 1 2 3 2
5 0.0 1 0 4 5 4
6 0.0 1 1 1
7 0.0 0 3 2 0 6 11 6
8 0.0 0 3 3 3
9 0.0 0 2 3 2 7 3
10 0.0 0 2 2 4 2
11 0.0 0 2 0 5 3 1 4 0 10 25 10
12 0.0 0 2 2 2
13 0.0 0 1 4 1 2 5 4 8 5 0 12 42 12
14 0.0 0 1 3 3 3 10 3
15 0.0 0 1 3 4 3
...
A299989: RETRACTED: Irregular triangle read by rows in which row n contains significant digits after the radix point for unit fractions 1/n expanded in primorial base.
22 February 2018 {0, 1, 0, 2, 0, 1, 10, 0, 1, 4, 0, 1, 0, 0, 17, 6, 0, 0, 15, 0, 0, 13, 14, 0, 0, 12, 0, 0, 10, 38, 1440, …}
By convention, row n = 1 contains {0}.
Length of row n = A139169(n).
Length of row prime(i) = i.
This sequence is retracted due to an error in my Mathematica code that led me to fail to prove the implicit declaration of the sequence: all integers n have terminating expansions in primorial base. This is false:
Unit fractions of many composite numbers 1/c do not terminate. A factorial-base representation, 1/c with c not squarefree can terminate for we have other “places” divisible by the square, cube, etc. of a given prime p, since all positive integers serve as a place in factorial base. This is not so in primorial base, for we have only one instance of a given prime. Therefore, 1/4 does not terminate in primorial base, but 1/6 does, since it is squarefree. Primorial base does not function like a “normal” number base that has additional “copies” of prime factors p in each place, therefore, if a fraction has multiplicity of any prime divisor in the denominator, it will be mixed-recurrent, like decimal 1/6 or 1/14. Factorial base is special since, not quite akin to “normal” bases, it does have “copies” of all prime factors somewhere further rightward of the radix point.
A plausible sequence can be written, mapping the primorial-base algorithm to unit fractions of squarefree numbers 1/A005117(n). It may be interesting to determine whether any fully recurrent fractions exist in primorial base; it would seem we only have mixed recurrent. Therefore, any non-squarefree number c “reads” like the unit fraction expansion of a semitotative in primorial base.
A299990: V0054: a(n) = A243822(n) − A000005(n).
25 February 2018 {−1, −2, −2, −3, −2, −3, −2, −4, −3, −2, −2, −4, −2, −2, −3, −5, −2, −2, −2, −4, −3, −1, −2, −5, −3, −1, −4, −4, −2, 2, −2, −6, −2, 0, …}
See Examination of the relationships of the species of numbers enumerated in A010846.
Since A010846(n) = A000005(n) + A243822(n), this sequence examines the balance of the two components among “regular” numbers.
Value of a(n) is generally less frequently negative as n increases.
a(1) = −1.
For primes p, a(p) = −2 since 1 | p and the cototient is restricted to the divisor p.
For perfect prime powers pe, a(pe) = −(e + 1), since all m < pe in the cototient of pe that do not have a prime factor q coprime to pe are powers pk with 1 < pk ≤ pe; all such pk divide pe.
Generally for n with A001221(n) = 1, a(n) = −1 × A000005(n), since the cototient is restricted to divisors, and in the case of pe > 4, divisors and numbers in A272619(pe) not counted by A010846(pe).
For m ≥ 3, a(A002110(m)) is positive.
For m ≥ 9, a(A244052(m)) is positive.
Some values of a(n) and related sequences:
n a(n) A010846(n) A243822(n) A000005(n) A272618(n)
----------------------------------------------------
1 -1 1 0 1 0
2 -2 2 0 2 0
3 -2 2 0 2 0
4 -3 3 0 3 0
5 -2 2 0 2 0
6 -3 5 1 4 {4}
7 -2 2 0 2 0
8 -4 4 0 4 0
9 -3 3 0 3 0
10 -2 6 2 4 {4,8}
11 -2 2 0 2 0
12 -4 8 2 6 {8,9}
...
30 2 18 10 8 {4,8,9,12,16,18,20,24,25,27}
...
34 0 8 4 4 {4,8,16,32}
...
A299991: V5401: Numbers n for which A243822(n) > A000005(n).
25 February 2018 {30, 42, 60, 66, 70, 74, 78, 82, 84, 86, 90, 94, 98, 102, 106, 110, 114, 118, 120, 122, 126, 130, 132, 134, 138, 140, 142, 146, 150, 154, 156, 158, 162, 165, …}
Composite numbers m have nondivisors k in the cototient such that k | ne with e > 1. These k appear in row n of A272618 and are enumerated by A243822(n). These nondivisors k are a kind of “regular” number along with divisors d of n; both are listed in row n of A162306 and are together enumerated by A010846(n).
This sequence lists numbers that have more nondivisors k in the cototient of n than divisors d.
This sequence contains all n for which A299990(n) is positive.
The smallest odd term is 165.
For m ≥ 3, A002110(m) is in a(n).
For m ≥ 9, A244052(m) is in a(n).
A299992: V5402: Composite n with A001221(n) > 1 for which A243822(n) < A000005(n).
26 February 2018 {6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 33, 35, 36, 39, 40, 44, 45, 48, 51, 52, …}
Composite numbers m have nondivisors k in the cototient such that k | n^e with e > 1. These k appear in row n of A272618 and are enumerated by A243822(n). These nondivisors k are a kind of "regular" number along with divisors d of n; both are listed in row n of A162306 and are together enumerated by A045763(n).
Primes p have 2 divisors {1, p}; these two numbers constitute the cototient of p: there are no nondivisors in the cototient.
Prime powers p^i have (i + 1) divisors; all smaller powers of the same prime p, i.e., p^j with 0 ≤ j ≤ i, also divide p^i. These numbers constitute the cototient of p^i; there are no nondivisors in the cototient.
Therefore, we can ignore cases where n has no nondivisors in the cototient, since they obviously have more divisors than nondivisors therein.
This sequence lists (composite) numbers n with omega(n) > 1 that have fewer nondivisors k in the cototient of n than divisors d.
The smallest odd term is 15.
The number m = 1001 is the smallest term with A001221(m) = 3. No term less than 36,000,000 has A001221(m) > 3.
The following terms m are the smallest to have A001222(m) = {2, 3, 4, ...}: {6, 12, 24, 48, 96, 192, 384, 1152, 2304, 4608, 13824, 27648, 55296, 110592, 331776, 663552, 1327104, 3981312, 7962624, 15925248, ...}.
Number of terms less than 10^k for 0 ≤ k ≤ 7: {0, 2, 44, 319, 2171, 15545, 119469, 969749}.
A300155: V5400: Numbers n for which A243822(n) = A000005(n).
26 February 2018 {34, 38, 46, 50, 54, 58, 62, 105, 249, 267, 268, 284, 291, 292, 303, 309, 316, 321, 324, 327, 332, 339, 356, 363, 381, 385, 388, 393, …}
Indices of zeros in A299990, i.e., A010846(n) − 2 × A000005(n) = 0.
Composite numbers m have nondivisors k in the cototient such that k | n^e with e > 1. These k appear in row n of A272618 and are enumerated by A243822(n). These nondivisors k are a kind of “regular” number along with divisors d of n; both are listed in row n of A162306 and are together enumerated by A045763(n). Divisors of n are listed in row n of A027750.
This sequence lists numbers that have an equal number of nondivisors k in the cototient of n as divisors d.
The smallest odd term is 105.
A300156: V5403: Indices of records in A299990.
26 February 2018 {1, 30, 42, 66, 78, 90, 102, 114, 138, 150, 210, 330, 390, 510, 570, 630, 870, 990, 1050, 1470, 1890, 2100, 2310, 2730, 3570, 3990, 4620, 5460, 6510, 6930, 8190, 9240, …}
See
Relationships between A244052, A294492, and A300156.
A010846(n) = A000005(n) + A243822(n).
Successive terms in this sequence represent increasing differences A243822(n) - A000005(n).
A000079 = records in −1 × A299990, since A243822(p^e) = 0 for e ≥ 0, n = 2^k sets records in A000005(n). The corresponding records are in A000027.
A300157: V5404: Records in A299990.
26 February 2018:
{−1, 2, 3, 6, 7, 8, 9, 10, 11, 17, 36, 45, 48, 54, 56, 67, 69, 76, 97, 118, 119, 120, 219, 231, …}
A010846(n) = A000005(n) + A243822(n).
A000079 = records in −1× A299990, since A243822(p^e) = 0 for e ≥ 0, n = 2^k sets records in A000005(n). The corresponding records are in A000027.
Number of terms less than 10^k with 0 <= k <= 7: {0, 1, 6, 18, 32, 51, 68, 96}.
A300858: V0046: a(n) = A243823(n) − A243822(n).
14 March 2018 {0, 0, 0, 0, 0, −1, 0, 1, 1, −1, 0, −1, 0, 1, 2, 4, 0, −1, 0, 3, 4, 3, 0, 3, 3, 5, 6, 7, 0, −5, 0, 11, …}
Consider numbers in the cototient of n, listed in row n of A121998. For composite n > 4, there are nondivisors m in the cototient, listed in row n of A133995. Of these m, there are two species. The first are m that divide n^e with integer e > 1, while the last do not divide n^e. These are listed in row n of A272618 and A272619, and counted by A243822(n) and A243823(n), respectively. This sequence is the difference between the latter and the former species of nondivisors in the cototient of n.
Since A045763(n) = A243822(n) + A243823(n), this sequence examines the balance of the two components among nondivisors in the cototient of n.
For positive n < 6 and for p prime, a(n) = a(p) = 0, thus a(A046022(n)) = 0.
For prime powers p^e, a(p^e) = A243823(p^e), since A243822(p^e) = 0, thus a(n) = A243823(n) for n in A000961.
Value of a(n) is generally nonnegative; a(n) is negative for n = {6, 10, 12, 18, 30}; a(30) = −5, but a(n) = −1 for the rest of the aforementioned numbers. These five numbers are a subset of A295523.
Some values of a(n) and related sequences:
n a(n) A243823(n) A243822(n) A272619(n) A272618(n)
-------------------------------------------------------------
1 0 0 0 - -
2 0 0 0 - -
3 0 0 0 - -
4 0 0 0 - -
5 0 0 0 - -
6 -1 0 1 - {4}
7 0 0 0 - -
8 1 1 0 {6} -
9 1 1 0 {6} -
10 -1 1 2 {6} {4,8}
11 0 0 0 - -
12 -1 1 2 {10} {8,9}
13 0 0 0 - -
14 1 3 2 {6,10,12} {4,8}
15 2 3 1 {6,10,12} {9}
16 4 4 0 {6,10,12,14} -
17 0 0 0 - -
18 -1 3 4 {10,14,15} {4,8,12,16}
19 0 0 0 - -
20 3 5 2 {6,12,14,15,18} {8,16}
...
A300859: V0042: Where records occur in A045763. “Highly Neutral Numbers”
15 March 2018 {1, 6, 10, 14, 18, 22, 26, 30, 36, 38, 42, 50, 54, 60, 66, 78, 84, 90, 102, 114, 120, …}
The cototient of n consists of numbers 1 < m ≤ n that are not coprime to n, i.e., gcd(m,n) > 1. These numbers have at least one prime divisor p that also divides n. The cototient of n contains the divisors d of n; the remaining nondivisors in the cototient of n are listed in A133995. The counting function of A133995 is A045763(n). There are two species of numbers in the nondivisor-cototient of n: those in row n of A272618, of which A243822(n) is counting function, and those in row n of A272619, of which A243823(n) is the counting function. The former species divides ne for integer e > 1, while the latter does not divide any integer power of n.
A045763(p) = 0 for p prime, therefore there are no primes in a(n).
Except for prime terms (i.e., 2), A002110 is a subset as primorials minimize the totient function. The divisor counting function is increasingly vanishingly small compared to the totient function for A002110(i) as i increases, and A002110(i) for 1 < i ≤ 9 is observed in a(n).
Conjectures based on 1255 terms of a(n) < 36,000,000:
1. There are no prime powers pe > 1 in a(n), i.e., the intersection of a(n) and A000961 is {1}.
2. A293555 is a subset of A300859. Numbers that have a lot of nondivisors m | ne with e > 1 (i.e., in row n of A272618 and counted by A243822(n)) tend to reduce the totient and increasingly have fewer divisors than highly composite numbers, widening the nondivisor-cototient.
3. A300156 is a subset of A300859. Numbers that have more nondivisors m | ne with e > 1 (i.e., in row n of A272618 and counted by A243822(n)) than divisors tend to reduce the totient and have fewer divisors than highly composite numbers (i.e., those n in A002182), widening the nondivisor-cototient.
Increasingly many terms k in A292867 also appear in a(n) as k increases. A292867 lists record-setters in A243823, which is the counting function of one of the two species of nondivisors in the cototient of n.
A300860: V0047: Indices of records in A300858.
14 March 2018 {1, 8, 15, 16, 26, 27, 28, 32, 44, 52, 56, 62, 64, 76, 80, 88, 96, 100, 104, 112, 122, …}
Consider numbers in the cototient of n, listed in row n of A121998. For composite n > 4, there are nondivisors m in the cototient, listed in row n of A133995. Of these m, there are two species. The first are m that divide n^e with integer e > 1, while the last do not divide ne. These are listed in row n of A272618 and A272619, and counted by A243822(n) and A243823(n), respectively. This sequence lists the record setters in the sequence A300858(n), which is a function that represents the difference between the latter and the former species of nondivisors in the cototient of n.
Odd terms m < 36,000,000: {1, 15, 27}.
Smallest term m with A001221(m) = {0, 1, 2, ..., 8} = {1, 8, 15, 246, 2010, 9870, 30030, 510510, 9699690} (the last 3 terms are in A002110).
Smallest term m with A001222(m) = {0, 2, 3, ..., 12} = {1, 15, 8, 16, 32, 64, 128, 256, 768, 1536, 7680, 53760, 3843840} (includes 2^e with 3 ≤ e ≤ 8). Note, A300858(p) for p prime = 0.
A300861: V0048: Records in A300858.
28 March 2018 {0, 1, 2, 4, 5, 6, 7, 11, 13, 17, 19, 21, 26, 27, 31, 35, 37, 40, 43, 47, 49, 51, 57, 66, …}
Consider numbers in the cototient of n, listed in row n of A121998. For composite n > 4, there are nondivisors m in the cototient, listed in row n of A133995. Of these m, there are two species. The first are m that divide ne with integer e > 1, while the last do not divide ne. These are listed in row n of A272618 and A272619, and counted by A243822(n) and A243823(n), respectively. This sequence lists the records in A300858, which is a function that represents the difference between the latter and the former species of nondivisors in the cototient of n.
A300914: V0043: Records in A045763.
15 March 2018 {0, 1, 3, 5, 7, 9, 11, 15, 16, 17, 23, 25, 29, 33, 39, 47, 49, 55, 63, 71, 73, 79, 81, …}
A301892: a(n) = A010846(A002182(n)). Number of regular k | me for highly composite m.
28 March 2018 {1, 2, 3, 5, 8, 11, 14, 15, 26, 36, 44, 49, 58, 76, 131, 156, 174, 206, 266, 308, 339, …}
We define an “n-regular” number as 1 ≤ m ≤ n such that m | ne with integer e ≥ 0. The divisor d is a special case of regular number m such that d | ne with e = 0 or e = 1. Regular numbers m can exceed n; we are concerned only with regulars m ≤ n herein.
Since highly composite numbers represent those numbers that set records in the divisor counting function A000005, and since the divisor is a special case of regular number, this sequence applies the "regular counting function" A010846 to terms in A002182.
Only 13 HCNs less than 36 × 106 are also “highly regular”, i.e., appear in A244052: these are in A168263.
Let “tier” t consist of all terms A002110(t) <= m < A002110(t + 1) in A244052, where all such m in tier t have A001221(m) = t. The intersection of A002182 and A244052 is finite, consisting of 13 terms: {1, 2, 4, 6, 12, 24, 60, 120, 180, 840, 1260, 1680, 27720}. All of these terms are also in A060735 and not in A288813, as the latter are squarefree and have “gaps” among prime divisors. This intersection has the following number of terms in the "tiers" 0 through 5 of A244052: {1, 2, 3, 3, 3, 1}. If we look at A060735 as a number triangle T(n,k) = k × A002110(n) with 1 ≤ k < prime(n + 1), the terms are:
(0,1)
(1,1), (1,2)
(2,1), (2,2), (2,4)
(3,2), (3,4), (3,6)
(4,4), (4,6), (4,8)
(5,12)
which correspond to A168263 = {1, 2, 4, 6, 12, 24, 60, 120, 180, 840, 1260, 1680, 27720}.
The largest HCN that is also highly regular is 27720, the 25th HCN and the 47th highly regular number.
Only 2 and 6 set records for the ratio A010846(n)/A000005(n).
A301893: Numbers m that set records for the ratio A010846(m)/A000005(m). This is the ratio of the regular counting function and the divisor counting function.
28 March 2018 {1, 6, 10, 18, 22, 30, 42, 66, 78, 102, 114, 138, 150, 174, 210, 330, 390, 510, 570, 690, …}
We define an “n-regular” number as 1 ≤ m ≤ n such that m | ne with integer e ≥ 0. The divisor d is a special case of regular number m such that d | ne with e = 0 or e = 1. Regular numbers m can exceed n; we are concerned only with regulars m ≤ n herein.
Since divisors are a special case of regular numbers, we examine those numbers m that set records for the ratio of the "regular counting function" A010846(m) and the divisor counting function A000005(m).
There are 2 nonsquarefree terms {18, 150} less than 36,000,000.
The sequence contains no numbers with ω(m) = 1. This is because all regular m divide pe, and since all the regulars of 1 also divide 1, no primes or prime powers greater than 1 appear in a(n).
The values of A000005(a(n)) are in A000079, i.e., powers 2e except e = 1.
Aside from the 2 nonsquarefree terms, many terms m are products of A002110(i) × pj, with j > i between some lower and upper bound outside of when m is in A002110. Example: 30 is in A002110; {42, 66, 78, 102, 114, 138, 174} are A002110(3) × pj with 2 ≤ j ≤ 8.
There are a few terms of the form A002110(i) × pj× pk, with i + 1 < j < k. In other words, there is a gap in the indices of the prime divisors between the 3rd and 2nd largest prime divisors, as well as one potentially between the 2nd and largest prime divisors. The smallest m of this type is 46410 = 2 × 3 × 5 × 7 × 13 × 17, followed by 51870 = 2 × 3 × 5 × 7 × 13 × 19.
Conjectures:
1. The only nonsquarefree terms are 18 and 150.
2. Primorials A002110(i) for i = 0 and i > 2 are in the sequence.
A301413: V1240: a(n) = A002182(n)/A002110(A108602(n)).
30 March 2018: {1, 1, 2, 1, 2, 4, 6, 8, 2, 4, 6, 8, 12, 24, 4, 6, 8, 12, 24, 36, 48, 72, 96, 120, 12, …}
Let m be a value in this sequence. The table below shows m*A002110(A108602(k)). Columns are A108602(k), rows are m whose products m*A002110(A108602(k)) appear in A002182 are in this sequence. Numbers in A002182 that also appear in A002201 are followed by (*).
A301414: V1241: Numbers k in A301413 such that k × A002110(m) is in A002182.
9 April 2018: {1, 2, 4, 6, 8, 12, 24, 36, 48, 72, 96, 120, 144, 216, 240, 288, 360, 480, 576, 720, 1080, …}
Given that highly composite numbers (HCNs) are products of primorials, we note the following:
1. The only odd term is 1.
2. The only primorials, i.e., terms in A002110, are {1, 2, 6}, consequently the only squares in A002182 are {1, 4, 36}.
3. The only terms in A000079 are {1, 2, 4, 8}. These produce {1, 2, 6}, {4, 12, 30}, {24, 120, 840}, and {48, 240, 1680}, in A002182 respectively.
4
. This sequence is a subset of A025487, which is a subset of A055932.
Also given that A002182 strictly increases, we note that i ≤ m ≤ j, integers, for which P = k × A002110(m) produces HCNs. As we increment m we increase the rank of the tensor of prime divisor power ranges and double the number of divisors. However, we may have another term P′= a * A002110(b) for a > k and b < (j + 1) such that P′ < P yet τ(P′) ≥ τ(P). This P′ is in A002182 and has increased τ by the lengthening of the power ranges for relatively small primes via some composite b instead of increasing the rank of the tensor. Since A002182 strictly increases, we have a limited range for m.
There are 19 terms also in A002182: 1, 2, 4, 6, 12, 24, 36, 48, 120, 240, 360, 720, 5040, 7560, 10080, 15120, 20160, 50400, 17297280.
Let n = A002110(m), and consider the ordered pair (n, k). In a plot of ordered pairs that produce m in A002182, we have the first terms of A002182 thus: (0,1), (1,1), (1,2), (2,1), (2,2), (2,4), (2,6), (2,8), (3,2), (3,4), (3,6), (3,8), (3,12), etc.
The table below plots the ordered pair (n,k) with the indices n of A002110 and k of A301414. This leads to a spectacular extended graph.
A301415: Number of terms m in A002110 such that A301414(k) × A002110(m) is in A002182.
9 April 2018 {3, 3, 3, 3, 3, 3, 4, 3, 3, 5, 3, 4, 4, 5, 5, 5, 4, 3, 4, 4, 4, 6, 3, 4, 5, 4, 3, 4, 3, …}
These are the number of distinct primorials
that yield a highly composite number when multiplied by A301414(k).
This sequence counts the terms in each column.
1 2 3 4 5 6 7 ...
+-----------------------------------------
0 | 1
1 | 2* 4
2 | 6* 12* 24 36 48
3 | 60* 120* 180 240 360* 720
4 | 840 1260 1680 2520* 5040*
5 | 27720 55440*
6 | 720720*
...
A301416: V1341: Numbers k in A301413 such that k × A002110 (m) is in A002201.
9 April 2018 {1, 2, 4, 12, 24, 48, 144, 720, 1440, 10080, 30240, 60480, 302400, 604800, …}
These are primitive values c that produce a superior highly composite number when multiplied by some primorial.
A304234: V1203: Superior highly composite numbers that are superabundant but not colossally abundant.
8 May 2018 {13967553600, 2248776129600, 65214507758400, 195643523275200, 12129898443062400, 448806242393308800, 18401055938125660800, …}
Numbers m in A002201 that are also in A004394 but not A004490.
Subset of A166981. Numbers in this sequence are in neither A224078 nor A304235.
There are 39 terms in this sequence.
The smallest term is 25 × 3² × 5 × A002110(8) or the product of A002110(k) with k = {1,1,1,2,3,8}.
The largest is 210 × 36 × 5³ × 7² × 11 × 13 × 17 × 19 × 23 × A002110(65) or the product of A002110(k) with k = {1,1,1,1,2,2,2,3,4,9,65}, a 144 digit decimal number.
"darkest blue" pixels in the chart below:
A304235: V1204: Colossally abundant numbers that are highly composite, but not superior highly composite.
8 May 2018 {160626866400, 9316358251200, 288807105787200, 2021649740510400, 224403121196654400, 9200527969062830400, …}
Numbers m in A004490 that are also in A002182 but not A002201. Subset of A166981. Numbers in this sequence are in neither A224078 nor A304234.
There are 32 terms in this sequence.
The smallest term is 24 × 3² × 5 × A002110(9) or the product of k = {1,1,2,3,9} in A002110.
The largest term is 29 × 35 × 5³ × 7² × 11 × 13 × 17 × 19 × 23 × A002110(66) or the product of A002110(k) with k = {1,1,1,1,2,2,3,4,9,66}, a 146 digit decimal number.
"darkest red" pixels in the chart above.
A304569: V0039: Triangle read by rows: T(n,k) = 1 if k | ne with e ≥ 0. T(n,k) = 0 otherwise (1 ≤ k ≤ n)
15 May 2018 {1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, …}.
Row n is characteristic function of numbers k ≤ n regular to n, i.e., k is in row n of A162306.
A010846(n) = total of row n in this sequence.
Row p for p prime begins and ends with 1, but otherwise contains zeros; it is equivalent to row p of A051731.
Row n for n such that ω(n) = 1 is the same as row n of A051731.
All other rows have additional 1s at positions in row n of A272618.
Table begins:
1;
1, 1;
1, 0, 1;
1, 1, 0, 1;
1, 0, 0, 0, 1;
1, 1, 1, 1, 0, 1;
1, 0, 0, 0, 0, 0, 1;
1, 1, 0, 1, 0, 0, 0, 1;
1, 0, 1, 0, 0, 0, 0, 0, 1;
1, 1, 0, 1, 1, 0, 0, 1, 0, 1;
...
A304570: Triangle read by rows: T(n,k) = 1 if k | ne with e > 1 and n mod k ≠ 0. T(n,k) = 0 otherwise (1 ≤ k ≤ n)
23 May 2018 {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, …}.
Row n is characteristic function of semidivisors of n.
T(n,k) = 1 iff A051731(n,k) = 0 but A304569(n,k) = 1; T(n,k) = 0 otherwise. This sequence contains 1 where 1 appears in row n of A304569 but not in same row of A051731.
Row n of A272618 contains indices of 1 in this sequence.
A243822(n) = total of row n in this sequence.
Row n such that ω(n) = 1 contains zeros.
Table begins:
0;
0, 0;
0, 0, 0;
0, 0, 0, 0;
0, 0, 0, 0, 0;
0, 0, 0, 1, 0, 0;
0, 0, 0, 0, 0, 0, 0;
0, 0, 0, 0, 0, 0, 0, 0;
0, 0, 0, 0, 0, 0, 0, 0, 0;
0, 0, 0, 1, 0, 0, 0, 1, 0, 0;
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;
0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0;
...
A304571: V0049: Triangle read by rows: T(n,k) = 1 if gcd(n,k) > 1 and n mod k ≠ 0.
23 May 2018 {, …}.
Row n is characteristic function of k ≤ n “neutral” to n.
T(n,k) = 1 iff both A051731(n,k) = 0 and A054521(n,k) = 0; T(n,k) = 0 otherwise.
This sequence contains 1 where 1 appears in row n of A304570 or A304572.
Row n of A133995 contains indices of 1 in this sequence.
A045763(n) = total of row n in this sequence.
Row p for p prime begins and ends with 1, but otherwise contains zeros; it is equivalent to row p of A051731.
Row n for n such that ω(n) = 1 contains only zeros; all other rows have at least one 1.
T(n,k) = 0 for k prime.
Table begins:
0;
0, 0;
0, 0, 0;
0, 0, 0, 0;
0, 0, 0, 0, 0;
0, 0, 0, 1, 0, 0;
0, 0, 0, 0, 0, 0, 0;
0, 0, 0, 0, 0, 1, 0, 0;
0, 0, 0, 0, 0, 0, 0, 0, 0;
0, 0, 0, 1, 0, 1, 0, 1, 0, 0;
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;
0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0;
...
A304572: V0069: Triangle read by rows: T(n,k) = 1 if k does not divide ne, positive nonzero integers, and gcd(n,k) > 1.
23 May 2018 {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, …}.
Row n is characteristic function of semitotatives of n.
T(n,k) = 1 iff both A304569(n,k) = 0 and A054521(n,k) = 0; T(n,k) = 0 otherwise.
This sequence contains 1 where 1 appears in row n of A304571 but not A304569.
Row n of A272619 contains indices of 1 in this sequence.
A243823(n) = total of row n in this sequence.
Rows n for n prime and n ≤ 6 contain only zeros; all other rows have at least one 1.
T(n,k) = 0 for k prime.
Table begins:
0;
0, 0;
0, 0, 0;
0, 0, 0, 0;
0, 0, 0, 0, 0;
0, 0, 0, 0, 0, 0;
0, 0, 0, 0, 0, 0, 0;
0, 0, 0, 0, 0, 1, 0, 0;
0, 0, 0, 0, 0, 1, 0, 0, 0;
0, 0, 0, 0, 0, 1, 0, 0, 0, 0;
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;
0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0;
...
A304886: Irregular triangle where row n contains indices k where the product of A002110(k) = A025487(n).
23 May 2018 {0, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 3, …}
Row n consists of terms k such that A025487(n) = the product of primorials pk#, the k in row n written least to greatest k.
For m = A025487(n) in A000079 (i.e., m is an integer power of 2), row n contains A000079(m) 1s.
For m = A025487(n) in A002110 (i.e., m is a primorial) row n contains a single term k that is the index of m in A002110.
Triangle begins as in rightmost column. Note that the maximum value in the rightmost column is A061395(n).
n A025487(n) Row n
--------------------------------
1 1 0
2 2 1
3 4 1,1
4 6 2
5 8 1,1,1
6 12 1,2
7 16 1,1,1,1
8 24 1,1,2
9 30 3
10 32 1,1,1,1,1
11 36 2,2
12 48 1,1,1,2
13 60 1,3
14 64 1,1,1,1,1,1
15 72 1,2,2
16 96 1,1,1,1,2
17 120 1,1,3
18 128 1,1,1,1,1,1,1
19 144 1,1,2,2
20 180 2,3
...
A305025: V1221: a(n) = A001221(A004394(n)).
30 June 2018 {0, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, …}
Number of distinct prime factors of superabundant numbers.
Analogous to A108602 (which instead pertains to A002182, the highly composite numbers).
a(23) = 5 while A108602(23) = 4; 23 is the smallest index where this sequence differs from A108602.
A305056: V1245: a(n) = A004394(n)/A002110(A001221(A004394(n))).
1 July 2018 {1, 1, 2, 1, 2, 4, 6, 8, 2, 4, 6, 8, 12, 24, 4, 6, 8, 12, 24, 48, 72, 120, …}
This sequence is analogous to A301413, which pertains to A002182.
Since A002182(20) = 7560 is not in A004394, a(20) ≠ A301413(20), i.e., the former is 36, the latter 48. (The number 36 is not in this sequence, but is in A301413.)
A004394(50) = 120 × A002110(8) is the smallest number in A004394 but not in A002182; in A004394 we have 120 × A002110(m) for 4 ≤ m ≤ 8, while in A002110 we have 120 × A002110(m) for 4 ≤ m ≤ 7. Therefore this sequence has one more instance of 120 (= a(50)) than exists in A301413.
Let m be a value in this sequence. The table below shows m × A002110(A001221(A004394(k))). Columns are A001221(A004394(k)), rows are m whose products m × A002110(A001221(A004394(k))) appear in A004394 are in this sequence. Numbers in A004394 that also appear in A004490 are followed by (*).
0 1 2 3 4 5 6 ...
+----------------------------------------------------
1 | 1 2* 6*
2 | 4 12* 60*
4 | 24 120* 840
6 | 36 180 1260
8 | 48 240 1680
12 | 360* 2520* 27720
24 | 720 5040* 55440* 720720*
Up to this point, the graph of this sequence and that of A301413 are identical; the asterisks pertain to numbers in A002201 in the case of A301413, but all the numbers on the graph are found in both A004490 and A002201, i.e., in A224078. The next two rows of the graph of A301413:
0 1 2 3 4 5 6 ...
+----------------------------------------------------
36 | 7560 83160 1081080
48 | 10080 110880 1441440*
...
but this sequence does not have row m = 36, as {7560, 83160, 1081080} are not in A004394.
A316990: Smallest exponent m of n such that A289280(n) | nm.
28 July 2018 {2, 2, 2, 2, 3, 2, 2, 2, 4, 2, 2, 2, 4, 2, 2, 2, 3, 2, 2, 3, 5, 2, 3, 2, 5, …}
Consider the least k > n such that k | nm for m > 1. (We note that k cannot divide n if k exceeds n.) Values of k appear in A289280, while this sequence lists values of m.
If row n of A162306 were extended to include terms greater than n, A289280(n) would be the first term to follow those already in the row.
a(n) = 2 for n with ω(n) = 1. In other words A289280(n) | n² for n = pe with one distinct prime divisor, since A289280(pe) = p(e+1).
First indices of {2, 3, 4, 5, ..., m} are {2, 6, 10, 22, 34, 74, 134, 262, 514, 1042, 2062, 4106, 8198, 16418, 32822, 65542, ...}, i.e., the least even squarefree semiprime s > 2(m − 1) for m > 2. This is because 2 is the smallest prime, and minimal multiplicity of 2 increases a(n) most efficiently. Let n = Product(pe) and A289280(n) = Product(pd), knowing there may be different values of p. a(n) = max(ceiling(d/e)) for d and e that pertain to the same prime p. Examples: for n = 10 = 2×5, A289280(10) = 16 = 24. Thus we are concerned with the ratio 4/1, and a(10) = 4. For n = 12 = 2²×3 we have A289280(12) = 16; here we have the ratio 4/2 = 2. The greater multiplicity of 2 reduces a(n) for n = 12.
A316991: Numbers m such that 1 < gcd(m, 14) < m and m does not divide 14e for e ≥ 0. (R14: Numbers m semicoprime to 2 and 7).
2 August 2018 {6, 10, 12, 18, 20, 21, 22, 24, 26, 30, 34, 35, 36, 38, 40, 42, 44, 46, 48, 50, …}
Complement of A000027 and union of A003591 and A162699.
Analogous to A081062 and A105115 that apply to A120944(1) and A120944(2), respectively.
This sequence applies to A120944(3) = 14.
A316992: Numbers m such that 1 < gcd(m, 15) < m and m does not divide 15e for e ≥ 0. (R15: Numbers m semicoprime to 3 and 5).
2 August 2018 {6, 10, 12, 18, 20, 21, 22, 24, 26, 30, 34, 35, 36, 38, 40, 42, 44, 46, 48, 50, …}
Complement of A000027 and union of A003593 and A229829.
Analogous to A081062 and A105115 that apply to A120944(1)=6 and A120944(2)=10, respectively.
This sequence applies to A120944(4) = 15.
A321223: a(n) is the number of recursively self-conjugate partitions of n.
31 October 2018 {1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, …}
A recursively self-conjugate partition L has a conjugate L* = L. Further, elimination of the Durfee square and leg (conjugate with the arm) to leave the arm L1. L1 likewise has conjugate L1* = L1. We continue taking the arm, eliminating the new Durfee square and leg in this manner until the entire partition is processed and all arms are self-conjugate.
We can define a recursively self-conjugate partition L by placing a series S of squares sk in position k, whose side-lengths decrease as k increases, in the following manner. We place the first square in the upper left corner, then set 2(k − 1) squares sk in all places wherein we have bounds by axis or previous square to the left and top. Thereby we can abbreviate all recursively self-conjugate partitions L by S(L). For example, (5,4,4,4,1) = {4,1}, and (10,9,8,7,6,5,4,3,2,1) = {5,3,1,1}. (See Keith 2011 page 9 Fig. 3.)
A190900 = positions of 0 in a(n).
A322156: Irregular triangle where row n includes all decreasing sequences S = {k0 = n, k1, k2, ..., km} in reverse lexical order such that the sum of subsequent terms kj for all i < j ≤ m does not exceed any ki.
11 December 2018 {1, 1, 1, 2, 2, 1, 2, 1, 1, 2, 2, 3, 3, 1, 3, 1, 1, 3, 2, 3, 2, 1, 3, 3, 4, 4, 1, 4, 1, 1, 4, 2, 4, 2, 1, 4, 2, 1, 1, 4, 2, 2, 4, 3, 4, 3, 1, 4, 4, …}
See diagram.
Algorithm:
Let S be a sequence starting with n. Let k be the index of a term in S, with n at position k = 0. Let Sr be the r-th sequence in row n.
Starting with S1 = {n}, we either (A) append a 1 to the left of Sr, or (B) we drop the most recently-appended term S(k) and increment the rightmost term (k − 1).
By default we execute (A) and test according to the following. Consider the reversed accumulation A(r + 1) = Sum(reverse(S(r + 1))) = Sum(km, k(m − 1), …, k2, k1). If Sr − A(r + 1) contains nothing less than 0, then S(k + 1) is retained, else we execute (B).
We end after k1 = n, since otherwise we would enter an endless loop that also increments k0 ad infinitum.
The first sequence S in row n is {n} while the last is {n, n}.
All rows n contain {{n}, {n, 1}, {n, n}}.
Only one repeated term k may appear at the end of any S in row n.
The longest possible sequence S in row n has 2 + floor(log2(n)) terms = 2 + A113473(n).
The sequence S describes unique integer partitions L that are recursively symmetrical.
Example: We can convert S = {4, 2, 1} into the partition (7, 6, 5, 4, 3, 2, 1), a partition of N = 28. We set a 4× Durfee square with its upper-left corner at origin. Then we set 2k = 2¹ = 2 2× squares with its upper-left corner in any coordinate bounded at left and top by either a previously-lain square or an axis. Finally, we set 2² = 4 1× squares as above once again. We obtain a Ferrer diagram as below, with the k marked, i.e., the 1st term 4×, the 2nd term 2×, the 3rd term 1× squares:
0 0 0 0 1 1 2
0 0 0 0 1 1
0 0 0 0 2
0 0 0 0
1 1 2
1 1
2
The resulting partition L is recursively self-conjugate; its arms are identical to its legs. We can eliminate the Durfee square and the other appendage and have a symmetrical partition L1 with Durfee square of k1 units, etc.
Were we to admit either more than 1 repeated k or a term such that Sk − A(k + 1) had differences less than 1, we would have overlapping squares in the Ferrer diagram. Such diagrams are generated by larger n and all resulting diagrams are unique given the described algorithm.
The sequences S in row n, converted into integer partitions L, sum to n² ≤ N ≤ 3 × n².
A322457: Irregular triangle: Row n contains numbers k that have recursively symmetrical partitions having Durfee square with side length n.
11 December 2018 {1, 3, 4, 6, 10, 12, 9, 11, 15, 17, 21, 27, 16, 18, 22, 24, 28, 34, 36, 38, 40, 48, 25, 27, 31, 33, 37, 43, 45, 47, 49, 55, …}
See diagram 1, diagram 2.
For all n, n² ≤ k ≤ 3 × n².
For n > 5, some k may have more than 1 recursively self-conjugate partitions in the same row. For example, k = 90 in row 6 has two recursively self-conjugate partitions (RSCPs) with Durfee square of 6: (12,12,12,9,9,9,6,6,6,3,3,3) and (12,11,11,11,11,7,6,5,5,5,5,1). These RSCPs can be defined by dendritically laying out squares in the series {6,3,3} and {6,5,1} respectively.
A323034: Where records occur in A321223.
2 January 2019 {1, 27, 103, 175, 198, 310, 411, 495, 627, 675, 720, 838, 880, 1008, …}
See diagram 1, diagram 2.
Numbers k that set records for the number m of recursively self-conjugate partitions (RSCPs).
1 is the only square in the sequence.
The graph of A321223 suggests there is a finite number of numbers k with a given number m of RSCPs. We know that A190900 (positive integers without RSCPs) is finite.
For index i ≤ 216, There are 6 squares in A321223, i.e., those of {1, 2, 3, 5, 8} that have just 1 RSCP; there are 120 non-squares 3 ≤ k ≤ 590 in A321223 that have m = 1 RSCP. In the same range, there are 127 numbers 27 ≤ k ≤ 830 in A321223 that have m = 2 RSCPs, and 142 numbers 103 ≤ k ≤ 1280 in A321223 that have m = 3 RSCPs. This sequence includes many of the first terms k of these finite sequences, all k having m RSCPs.
Examining the smallest 381 terms (i.e., all k < 216) and the graph of A321223, we observe the following:
1. a(3) = 103 and a(23) = 2011 are the only primes.
2. a(2) = 27 = 3³ and a(64) = 6561 = 38 are the only prime powers.
3. Numbers k such that k mod 3 = 2 are never in this sequence.
4.
Only k in {1, 103, 175, 310, 838, 880, 2011, 2416, 4531, 4720, 5872, 11248, 11632, 12400, 15136, 16081, 19696, 20464, 29296, 40816, 51568, 52336} are congruent to 1 (mod 3); this of course includes both primes 103 and 2011.
Conjecture:
k in this sequence is never congruent to 2 (mod 3).
RSCPs of the first 3 terms:
a(1) = 1: (1).
a(2) = 27: (6,6,6,3,3,3), (6,5,5,5,5,1).
a(3) = 103: (13,13,13,10,10,10,7,6,6,6,3,3,3), (13,12,12,12,12,8,7,6,5,5,5,5,1), (13,12,12,10,9,9,9,9,9,4,3,3,1)
A323035: Records in A321223.
4 January 2019 {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 23, 25, 26, 28, 29, …}
Same diagrams as A323034.
n = 1014 is the smallest n to have 15 recursively self-conjugate partitions (RSCPs); n = 1191 is the smallest n to have 17 RSCPs. 16 is not in the sequence because the smallest n to have 16 RCSPs is 1200; 1200 exceeds 1191, and thus 16 does not set a record in A321223.
A306737: Irregular triangle where row n is a list of indices in A002110 with multiplicity whose product is A002182(n).
6 March 2019 {0, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 2, 1, 1, 1, 2, 1, 3, 1, 1, 3, 2, 3, 1, 1, 1, 3, …}
The sequence is of general interest because each highly composite number A002182(n) can be expressed as a product of primorials in A002110. This is tantamount to taking the conjugate of the chart of the prime powers that produce HCNs:
2520 = 2³ × 3² × 5 × 7
======================
2 --> 2
2 3 --> 6
×2 ×3 5 7 --> × 210
-- -- -- -- --> -----
8 × 9 × 5 × 7 --> 2310
Row 1 = {0} by convention.
Maximum value in row n = A001221(A002182(n)).
Row n in reverse order is the conjugate of the list of the multiplicities of the prime divisors of A002182(n).
Terms in the first rows n of this sequence, followed by the corresponding primorials whose product = A002182(n):
n T(n,k) A002110(T(n,k)) A002182(n)
-----------------------------------------------
1: 0; 1 = 1
2: 1; 2 = 2
3: 1, 1; 2 × 2 = 4
4: 2; 6 = 6
5: 1, 2; 2 × 6 = 12
6: 1, 1, 2; 2 × 2 × 6 = 24
7: 2, 2; 6 × 6 = 36
8: 1, 1, 1, 2; 2 × 2 × 2 × 6 = 48
9: 1, 3; 2 × 30 = 60
10: 1, 1, 3; 2 × 2 × 30 = 120
11: 2, 3; 6 × 30 = 180
12: 1, 1, 1, 3; 2 × 2 × 2 × 30 = 240
13: 1, 2, 3; 2 × 6 × 30 = 360
14: 1, 1, 2, 3; 2 × 2 × 6 × 30 = 720
15: 1, 1, 4; 2 × 2 × 210 = 840
...
A306802: Position of highly composite numbers in the sequence of products of primorials.
12 March 2019 {1, 2, 3, 4, 6, 8, 11, 12, 13, 17, 20, 24, 27, 34, 36, 43, 47, 55, 67, 77, 84, 95, …}
This sequence is of general interest since A025487 is the sequence of products of primorials (A002110), and we can state terms in A002182 as a product of primorials; not all numbers in A025487 are in A002182.
Indices of A002182 in A025487. All terms in A002182 are products of terms in A002110; A025487 lists products of terms in A002110.
The first 28 terms of this sequence and those of A293635 (Position of superabundant numbers in the sequence of products of primorials) are identical since the smallest 28 terms of A002182 and A004394 are the same.
A307056: Row n = digits of A025487(n) in primorial base.
21 March 2019 {1, 1, 0, 2, 0, 1, 0, 0, 1, 1, 0, 2, 0, 0, 2, 2, 0, 4, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, …}
This sequence is of general interest given A025487 is the sequence of products of primorials (A002110). This sequence sees terms in A025487 through the “lens” of the primorials as a mixed radix.
First rows of this sequence:
n A025487(n) Row n
-------------------------------
1 1 1
2 2 1, 0
3 4 2, 0
4 6 1, 0, 0
5 8 1, 1, 0
6 12 2, 0, 0
7 16 2, 2, 0
8 24 4, 0, 0
9 30 1, 0, 0, 0
10 32 1, 0, 1, 0
11 36 1, 1, 0, 0
12 48 1, 3, 0, 0
13 60 2, 0, 0, 0
14 64 2, 0, 2, 0
15 72 2, 2, 0, 0
16 96 3, 1, 0, 0
17 120 4, 0, 0, 0
18 128 4, 1, 1, 0
19 144 4, 4, 0, 0
20 180 6, 0, 0, 0
21 192 6, 2, 0, 0
22 210 1, 0, 0, 0, 0
...
A307113: Number of highly composite numbers (m in A002182) in the interval pk# ≤ m < p(k+1)#, where pi# = A002110(i).
25 March 2019 {1, 2, 3, 5, 6, 8, 10, 12, 13, 15, 14, 15, 17, 16, 16, 19, 17, 21, 19, 20, …}
Terms m in A002182 (highly composite numbers or HCNs) are products of primes p ≤ q, where q is the greatest prime factor of m. The primorial A002110(k) is the smallest number that is the product of the k smallest primes. This sequence partitions A002182 using terms in A002110.
n a(n) m such that A002110(n) <= m < A002110(n+1)
--------------------------------------------------------------------
0 1 1
1 2 2 4
2 3 6 12 24
3 5 36 48 60 120 180
4 6 240 360 720 840 1260 1680
5 8 2520 5040 7560 10080 15120 20160 25200 27720
The sequence has an interesting graph. I used Flammenkamp's 779,674 HCN dataset to generate 4150 terms:
A307133: T(n,m) = number of k ≤ A002110(n) such that A001222(k) = m, where k is a term in A025487.
26 March 2019 {1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 7, 9, 4, 1, 1, 11, 21, 15, 5, 1, 1, 14, 38, 36, 18, 6, 1, …}
Terms m in A025487 are products of pn# in A002110.
The primorial pn# = A002110(n) is the smallest number that is the product of the n smallest primes and is a subset of A025487.
T(n, 0) = T(n, n) = A000012(n).
T(n, 1) = A054850(n).
A098719(n) = sum of row n.
Row 3 = {1,4,3,1}. The terms k in A025487 such that k ≤ A002110(3) are {1, 2, 4, 6, 8, 12, 16, 24, 30}. Of these, 1 has 0 distinct prime divisors, 4 {2,4,8,16} have 1 distinct prime divisor, 3 {6,12,24} have 2 distinct prime divisors, and 1 {30} has 3 distinct prime divisors.
Triangle begins:
0: 1
1: 1 1
2: 1 2 1
3: 1 4 3 1
4: 1 7 9 4 1
5: 1 11 21 15 5 1
6: 1 14 38 36 18 6 1
7: 1 18 64 79 53 23 7 1
8: 1 23 97 148 122 63 26 7 1
9: 1 27 140 258 251 157 76 30 7 1
10: 1 32 196 425 480 349 195 89 33 8 1
11: 1 37 261 655 853 700 443 228 102 37 9 1
12: 1 42 340 975 1438 1323 928 533 268 119 41 11 1
...
A307107: a(n) = A025487(n)/A247451(n).
29 March 2019 {1, 1, 2, 1, 4, 2, 8, 4, 1, 16, 6, 8, 2, 32, 12, 16, 4, 64, 24, 6, 32, 1, 36, 8, 128, …}.
Ratio of A025487(n) and the largest primorial that divides A025487(n). The largest primorial that divides A025487(n) is pω(n)#.
If A025487(n) is a primorial p#, a(n) = 1.
a(n) is in A025487 by definition of that sequence as a sorted list of products of primorials.
Conjectures:
1. 1 is the most common value in this sequence even though it only pertains to primorials.
2. All terms in A025487 are in this sequence.
We can represent the prime divisors p with multiplicity of A025487(n) in a chart where the columns pertain to p and the rows multiplicity. In such a chart, A247451(n) is the longest row (marked by "O" below), and a(n) is the product of primes left over (marked by "X") when we eliminate the primes that produce A247451(n).
A025487(9) = 30 = a(9) × A247451(9)
= 1 * 30
1 O O O
2 3 5
A025487(27) = 360 = a(27) × A247451(27)
= 12 * 30
3 X
2 X X
1 O O O
2 3 5
A025487(183) = 166320 = a(183) × A247451(183)
= 72 * 2310
4 X
3 X X
2 X X
1 O O O O O
2 3 5 7 11
A307322: Irregular triangle where row n is a list of indices in A002110 with multiplicity whose product is A004394(n). (Analogous to A306737.)
2 April 2019 {0, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 2, 1, 1, 1, 2, 1, 3, 1, 1, 3, 2, 3, 1, 1, 1, 3, …}.
The first 52 terms of a(n) and A306737 are identical, since the first 19 terms of A002182 and A004394 are the same, and the first two terms of row 20 are the same. a(20) = 4,2,1,1,1, while A306737(20) = 4,2,2.
The sequence is of general interest because each superabundant number A004394(n) can be expressed as a product of primorials in A002110. This is tantamount to taking the conjugate of the chart of the prime powers that produce superabundant numbers:
2520 = 2³ × 3² × 5 × 7
======================
2 --> 2
2 3 --> 6
×2 ×3 5 7 --> × 210
-- -- -- -- --> -----
8 × 9 × 5 × 7 --> 2310
Row 1 = {0} by convention.
Maximum value in row n = A001221(A004394(n)).
Row n in reverse order is the conjugate of the list of the multiplicities of the prime divisors of A004394(n).
Terms in the first rows n of this sequence, followed by the corresponding primorials whose product = A004394(n):
n T(n,k) A002110(T(n,k)) A004394(n)
-----------------------------------------------
1: 0; 1 = 1
2: 1; 2 = 2
3: 1, 1; 2 × 2 = 4
4: 2; 6 = 6
5: 1, 2; 2 × 6 = 12
6: 1, 1, 2; 2 × 2 × 6 = 24
7: 2, 2; 6 × 6 = 36
8: 1, 1, 1, 2; 2 × 2 × 2 × 6 = 48
9: 1, 3; 2 × 30 = 60
10: 1, 1, 3; 2 × 2 × 30 = 120
11: 2, 3; 6 × 30 = 180
12: 1, 1, 1, 3; 2 × 2 × 2 × 30 = 240
13: 1, 2, 3; 2 × 6 × 30 = 360
14: 1, 1, 2, 3; 2 × 2 × 6 × 30 = 720
15: 1, 1, 4; 2 × 2 × 210 = 840
...
A307327: Number of superabundant numbers (m in A004394) in the interval pk# ≤ m < p(k+1)#, where pi# = A002110(i). (Analogous to A307113.)
2 April 2019 {1, 2, 3, 5, 6, 6, 5, 9, 8, 9, 8, 11, 12, 11, 11, 10, 12, 12, 11, 14, 15, 15, …}
Terms m in A004394 (superabundant numbers) are products of primes p ≤ q, where q is the greatest prime factor of m. The primorial A002110(k) is the smallest number that is the product of the k smallest primes. This sequence partitions A004394 using terms in A002110.
First terms {1, 2, 3, 5, 6} are the same as those of A307113, since the first 19 terms of A002182 and A004394 are identical.
n a(n) m such that A002110(n) <= m < A002110(n+1)
--------------------------------------------------------------------
0 1 1
1 2 2 4
2 3 6 12 24
3 5 36 48 60 120 180
4 6 240 360 720 840 1260 1680
5 6 2520 5040 10080 15120 25200 27720
6 5 55440 10080 166320 277200 332640
7 9 554400 665280 720720 1441440 2162160 3603600 4324320 7207200 8648640
The sequence has an interesting graph we can compare with that of A307113. We plot A307113 in blue, and a(n) in red. This sequence uses the 8436 terms of A004394 available in its b-file:
Compare this to the following, where the blue indicates terms m in A002182, red A004394, and grey/black terms m in both sequences. Let q# be the largest primorial that divides m, and let k = A301414(m) = m/q#. The axes x = k, while y = q#. From this graph it is clear that the superabundant numbers divisible by q# are generally smaller than the HCNs. More importantly, not all terms k in A301414 pertain to superabundant numbers. Hence we see fewer superabundant numbers than highly composite in each interval pk# ≤ m < p(k+1)#.
A306237: V2109: Primorial pn#/p(n − 1).
10 April 2019 {3, 10, 42, 330, 2730, 39270, 570570, 11741730, 281291010, 6915878970, 239378649510, 8222980095330, …}.
Easy sequence containing the zerocode pattern 1, i.e., in terms of A067255(a(n)), {x, 0, 1}, where x is a constant array of (n − 2) 1s. These numbers are highly regular (i.e., in A244052) and have totient ratios that are second-smallest for squarefree k such that gpf(k) = n.
Terms m in this sequence have the second-smallest totient ratio φ(m)/m for m with gpf(m) = prime(n). Only primorial pn# = A002110(n) has a smaller totient ratio (i.e., A038110(n)/A060753(n)). The chart below plots squarefree m at {π(gpf(m)), φ(m)/m}; the terms in this sequence are second from bottom in each column.
A307540: V2100: Irregular triangle T(n, k) such that squarefree m with gpf(m) = prime(n) in each row are arranged according to increasing values of φ(m)/m.
13 April 2019, {1, 2, 6, 3, 30, 10, 15, 5, 210, 42, 70, 14, 105, 21, 35, 7, 2310, 330, 462, 66, 770, 110, 154, 1155, 22, 165, 231, 33, 385, 55, 77, 11, 30030, …}.
Let gpf(m) = A006530(m) and let φ(m) = A000010(m) for m in A005117.
Row n contains m in A005117 such that A006530(m) = n, sorted such that φ(m)/m increases as k increases.
Let m be the squarefree kernel A007947(m') of m. We only consider squarefree m since φ(m)/m = φ(m' )/m'. Let prime p | n and prime q be a nondivisor of n.
Since m is squarefree, we might encode the multiplicities of its prime divisors in a positional notation M that is finite at n significant digits. For example, m = 42 can be encoded reverse(A067255(42)) = 1,0,1,1 = 71 × 50 × 31 × 21. It is necessary to reverse row m of A067255 (hereinafter simply A067255(m)) so as to preserve zeros in M = A067255(m) pertaining to small nondivisor primes q < p. The code M is a series of 0s and 1s since m is squarefree. Then it is clear that row n contains all m such that A067255(m) has n terms, and there are 2(n − 1) possible terms for n ≥ 1.
We may use an approach that generates the binary expansion of the range 2(n − 1) ≤ M ≤ 2n − 1, or we may append 1 to the reversed (n − 1)-tuples of {1, 0} to achieve codes M → m for each row n.
Originally it was thought that the codes M were in order of the latter algorithm, and we could avoid sorting. Observation shows that the m still require sorting by the function φ(m)/m indeed to be in increasing order in row n. Still, the latter approach is slightly more efficient than the former in generating the sequence.
For n > 0, row lengths = A000079(n − 1).
T(n, 1) = A002110(n) = pn#.
T(n, 2) = A306237(n) = pn#/prime(n − 1).
T(n, 2n − 1) = A006094(n).
T(n, n) = A000040(n) = prime(n).
Last even term in row n = A077017(n).
First odd term in row n = A070826(n).
The chart below plots squarefree m at {π(gpf(m)), φ(m)/m}; the terms in this sequence are read bottom to top in each column, proceeding left to right.
A307544: V2105: T(n,k) = A087207(A307540(n,k)). (Binary encoding of A307540)
19 April 2019, {0, 1, 3, 2, 7, 5, 6, 4, 15, 11, 13, 9, 14, 10, 12, 8, 31, 23, 27, 19, 29, 21, 25, 30, 17, 22, 26, 18, 28, 20, 24, 16, 63, …}.
Let gpf(m) = A006530(m) and let φ(m) = A000010(m) for m in A005117.
Row n contains m in A005117 such that A006530(m) = n, sorted such that φ(m)/m increases as k increases.
Let m be the squarefree kernel A007947(m') of m'. We only consider squarefree m since φ(m)/m = φ(m')/m'. Let prime p | n and prime q be a nondivisor of n.
Since m is squarefree, we might encode the multiplicities of its prime divisors in a positional notation M that is finite at n significant digits. For example, m = 42 can be encoded reverse(A067255(42)) = 1,0,1,1 = 71 × 50 × 31 × 21. It is necessary to reverse row m of A067255 (hereinafter simply A067255(m)) so as to preserve zeros in M = A067255(m) pertaining to small nondivisor primes q < p. The code M is a series of 0s and 1s since m is squarefree. Then it is clear that row n contains all m such that A067255(m) has n terms, and there are 2(n − 1) possible terms for n ≥ 1.
We may use an approach that generates the binary expansion of the range 2(n − 1) < M < 2n − 1, or we may append 1 to the reversed (n − 1)-tuples of {1, 0} (as A059894) to achieve codes M → m for each row n.
Originally it was thought that the codes M were in order of the latter algorithm, and we could avoid sorting. Observation shows that the m still require sorting by the function φ(m)/m indeed to be in increasing order in row n. Still, the latter approach is slightly more efficient than the former in generating the sequence.
This sequence interprets the code M as a binary value. The sequence is a permutation of the natural numbers since the ratio φ(m)/m is unique for squarefree m.
This sequence and A059894 are identical for 1 ≤ n ≤ 23.
Numbers of terms in rows n of this sequence and A059894 (partitioned by powers of 2) that are coincident: 1, 2, 4, 8, 14, 14, 10, 26, 14, 20, 10, 16, 22, 12, 18, 18, 16, 14, 18, 18, 18, 14, 16, ...}.
The graphs of this sequence and A059894 are similar.
The graph of this sequence feature squares of size 2(j − 1) at (x,y) = (h,h) where h = 2 j, integers, that have π-radian rotational symmetry.
Below, we show the graphs of A059894 (blue) and A307544 (red). Green points are those where A059894 and A307544 agree.
A325236: V2103: Squarefree k such that φ(k)/k − ½ is positive and minimal for k with gpf(k) = prime(n).
19 April 2019, {1, 2, 3, 15, 21, 231, 273, 255, 285, 167739, 56751695, 7599867, 3829070245, 567641679, 510795753, 39169969059, …}.
Let gpf(m) = A006530(m) and let φ(m) = A000010(m) for m in A005117. There are 2(n − 1) numbers k with gpf(k) = prime(n), since we can only either have pi0 or pi1 where pi | k and i ≤ n. For example, for n = 2, there are only 2 squarefree numbers k with prime(2) = 3 as greatest prime factor. These are 3 = 20 × 31, and 6 = 21 × 31. We observe that we can write multiplicities of the primes as A067255(k), and thus for the example derive 3 = “0,1” and 6 = “1,1”. Thus for n = 3, we have 5 = “0,0,1”, 15 = “0,1,1”, 10 = “1,0,1”, and 30 = “1,1,1”. This establishes the possible values of k with respect to n. We choose the value of k in n for which φ(k)/k − ½ is positive and minimal.
We know that prime k (in A000040) have φ(k)/k = A006093(n)/A000040(n) and represent maxima in n. We likewise know primorials k (in A002110) have φ(k)/k = A038110(n)/A060753(n) and represent minima in n. This sequence shows squarefree numbers k with gpf(k) = n such that their value φ(k)/k is closest to but more than ½.
The terms in this sequence are read just above the centerline in the chart below.
A325237: V2107: Squarefree k such that ½ − φ(k)/k is positive and minimal for k with gpf(k) = prime(n).
19 April 2019, {2, 6, 10, 105, 165, 195, 4641, 5187, 5313, 266133, 8870433, 3068957045, 11063481, 10164297, 667797009, 909411789, 32221169781185, …}.
Let gpf(m) = A006530(m) and let φ(m) = A000010(m) for m in A005117. There are 2(n − 1) numbers k with gpf(k) = prime(n), since we can only either have pi0 or pi1 where pi | k and i ≤ n. For example, for n = 2, there are only 2 squarefree numbers k with prime(2) = 3 as greatest prime factor. These are 3 = 20 × 31, and 6 = 21 × 31. We observe that we can write multiplicities of the primes as A067255(k), and thus for the example derive 3 = “0,1” and 6 = “1,1”. Thus for n = 3, we have 5 = “0,0,1”, 15 = “0,1,1”, 10 = “1,0,1”, and 30 = “1,1,1”. This establishes the possible values of k with respect to n. We choose the value of k in n for which φ(k)/k − ½ is positive and minimal.
We know that prime k (in A000040) have φ(k)/k = A006093(n)/A000040(n) and represent maxima in n. We likewise know primorials k (in A002110) have φ(k)/k = A038110(n)/A060753(n) and represent minima in n. This sequence shows squarefree numbers k with gpf(k) = n such that their value φ(k)/k is closest to but less than ½.
a(0) = 0 by convention, since there are no terms k such that φ(k)/k < ½ with n = 0.
The terms in this sequence are read just below the centerline in the chart below.
A307805: a(n) = first position of prime(n) in A023503.
29 April 2019 {2, 4, 5, 10, 9, 16, 27, 43, 15, 17, 64, 35, 23, 40, 61, 28, 127, 73, 57, 104, 62, …}.
A023503(n) = A006530(A006093(n)). Apparent permutation of A071349(n) apart from A071349(1) = 1.
Let i = a(n). Sorting prime(n) in order of increasing i yields A112037 = {2, 3, 5, 11, 7, 23, 13, 29, 41, ...}. The product of the first j terms of A112037 = A071350(j).
The chart below shows the multiplicity notation MN(pn − 1) = A067255(A006093(n)) in reverse, with the multiplicity of gpf(pn − 1) highlighted. We observe the first prime at n = 2, the second at n = 4, the third at n = 5, etc.
1 .
2 1*
3 2
4 1 1*
5 1 . 1*
6 2 1
7 4
8 1 2
9 1 . . . 1*
10 2 . . 1*
11 1 1 1
12 2 2
13 3 . 1
14 1 1 . 1
15 1 . . . . . . . 1*
16 2 . . . . 1*
17 1 . . . . . . . . 1*
...
A306999: V6504: Numbers m such that 1 < gcd(m, 21) < m and m does not divide 21e for e ≥ 0
.
22 August 2019 {6, 12, 14, 15, 18, 24, 28, 30, 33, 35, 36, 39, 42, 45, 48, 51, 54, 56, 57, 60, …}.
Complement of A000027 and the union of A003594 and A160545.
Analogous to A081062 and A105115 regarding terms 1 and 2 of A120944, respectively. This sequence applies to A120944(5) = 21.
Numbers m in S21 that are semicoprime to numbers k that have 3 and 7 and no other primes as distinct prime divisors.
A307589: V6505: Numbers m such that 1 < gcd(m, 35) < m and m does not divide 35e for e ≥ 0
.
22 August 2019 {6, 12, 14, 15, 18, 24, 28, 30, 33, 35, 36, 39, 42, 45, 48, 51, 54, 56, 57, 60, …}.
Complement of A000027 and the union of A003595 and A235933.
Analogous to A081062 and A105115 for terms 1 and 2 of A120944. This sequence applies to A120944(6) = 35.
Numbers m in S35 that are semicoprime to numbers k that have 5 and 7 and no other primes as distinct prime divisors.
A327893: Minesweeper sequence of positive integers arranged in a hexagonal spiral.
9 October 2019 {4, −1, −1, 3, −1, 3, −1, 3, 2, 4, −1, 3, −1, 3, 2, 2, −1, 3, −1, 2, 0, 2, −1, 3, 1, 2, …}
Place positive integers on 2-d grid starting with 1 in the center and continue along a hexagonal spiral. Replace primes by −1 and nonprimes by the number of primes in adjacent grid cells around them. n is replaced by a(n). This sequence treats prime numbers as “mines” and fills gaps according to classical Minesweeper game.
The largest term in the sequence is 4 since 1 is surrounded by 3 odd numbers {3, 5, 7} and the only even prime. Additionally, the pattern of odd and even numbers appears in alternating rows oriented in a triangular symmetry such that no other number has more than four odd numbers.
A330136: V6500: Numbers m such that 1 < gcd(m, 6) < m and m does not divide 6e for e ≥ 0
.
2 December 2019 {10, 14, 15, 20, 21, 22, 26, 28, 30, 33, 34, 38, 39, 40, 42, 44, 45, 46, 50, 51, 52, 56, 57, 58, 60, …}.
Numbers m that are neither 3-smooth nor reduced residues mod 6. Such numbers m have at least 1 prime factor p ≤ 3 and at least 1 prime factor q > 3. Complement of the union of A003586 and A007310. Analogous to A105115 for A120944(2) = 10. This sequence applies to A120944(1) = 6 = A002110(1).
The only composite n in A024619 for which n < A096014(n) is 6. Let n be a composite that is not a prime power (i.e., in A024619), let p = lpf(n) = A020639(n), and let q = A053669(n) be the smallest prime that does not divide n. We observe that A096014(n) = A020639(n) × A053669(n) = pq. Such n with n < pq must minimize one factor while maximizing the other. The prime p is minimum when n is even, and q is greatest when n is the product pk# of the smallest k primes, i.e., when n is in A002110. Alternatively, q is minimum when n is odd, however, n > 2p since n is the product of at least two distinct odd primes. Since pk# greatly increases as k increments, while A053669(pk#) = p(k + 1), and observing that A096014(30) = 2 × 7 = 14, the only composite n in A024619 such that n < pq is 6.
Numbers m in S6 that are semicoprime to numbers k that have 2 and 3 and no other primes as distinct prime divisors.
A330137: V6600: Numbers m such that 1 < gcd(m, 30) < m and m does not divide 30e for e ≥ 0
.
2 December 2019 {14, 21, 22, 26, 28, 33, 34, 35, 38, 39, 42, 44, 46, 51, 52, 55, 56, 57, 58, 62, …}.
Numbers m that are neither 5-smooth nor reduced residues mod 30. Such numbers m have at least 1 prime factor p ≤ 5 and at least 1 prime factor q > 5. Complement of the union of A007775 and A051037.
Analogous to A105115 for A120944(2) = 10. This sequence applies to the second primorial in A120944, i.e., 30 = A002110(2).
Numbers m in S30 that are semicoprime to numbers k that have 2, 3, and 5 and no other primes as distinct prime divisors.
A330781: Numbers m that have recursively self-conjugate prime signatures.
15 January 2019 {1, 2, 12, 36, 360, 27000, 75600, 378000, 1587600, 174636000, 1944810000, 5762988000, …}.
Let m be a product of a primorial, listed by A025487.
Consider the standard form prime power decomposition of m = Π(pe), where prime p | m (listed from smallest to largest p), and e is the largest multiplicity of p such that pe | m (which we shall hereinafter simply call “multiplicity”).
Products of primorials have a list L of multiplicities in a strictly decreasing arrangement.
A recursively self-conjugate L has a conjugate L* = L. Further, elimination of the Durfee square and leg (conjugate with the arm) to leave the arm L_1. L_1 likewise has conjugate L_1* = L_1. We continue taking the arm, eliminating the new Durfee square and leg in this manner until the entire list L is processed and all arms are self-conjugate.
This sequence is a subsequence of A181825 (m with self-conjugate prime signatures).
Subsequences of this sequence include A006939 and A181555.
This sequence can be produced by a similar algorithm that pertains to recursively self-conjugate integer partitions at A322156.
A331553: Irregular triangle T(n, k) = n − A115722(n, k)².
20 January 2020 {0, 0, 1, 1, 2, 2, 2, 3, 3, 0, 3, 3, 4, 4, 1, 4, 1, 4, 4, 5, 5, 2, 5, 2, 2, 5, 2, 2, 5, 5, …}.
Let P be an integer partition of n, and let D be the Durfee square of P with side length s, thus area s². We borrow the term "square excess" from A053186(n), which is simply the difference n − floor(sqrt(n)). This sequence lists the "Durfee square excess" of P = n − s² for all partitions P of n in reverse lexicographic order.
Zero appears in row n for n that are perfect squares. Let r = sqrt(n). For perfect square n, there exists a partition of n that consists of a run of r parts that are each r themselves; e.g., for n = 4, we have {2, 2}, for n = 9, we have {3, 3, 3}. It is clear through the Ferrers diagram of these partitions that they are equivalent to their Durfee square, thus n − s² = 0.
Since the partitions of any n contain Durfee squares in the range of 1 ≤ s ≤ floor(sqrt(n)) (with perfect square n also including k = 0), the distinct Durfee square excesses must be the differences n − s² for 1 ≤ s ≤ floor(sqrt(n)).
Table begins:
1 0;
2 1, 1;
3 2, 2, 2;
4 3, 3, 0, 3, 3;
5 4, 4, 1, 4, 1, 4, 4;
6 5, 5, 2, 5, 2, 2, 5, 2, 2, 5, 5;
7 6, 6, 3, 6, 3, 3, 6, 3, 3, 3, 6, 3, 3, 6, 6;
8 7, 7, 4, 7, 4, 4, 7, 4, 4, 4, 4, 7, 4, 4, 4, 4, 7, 4, 4, 4, 7, 7;
9 8, 8, 5, 8, 5, 5, 8, 5, 5, 5, 5, 8, 5, 5, 5, 5, 5, 8, 0, 5, 5, 5, 5, 5, 8, 5, 5, 5, 8, 8;
…
A331118: Irregular table where row n lists primitive run lengths L of numbers m < A002110(n) that are in the cototient of A002110(n).
10 January 2020 {4, 2, 4, 6, 2, 4, 6, 8, 10, 2, 4, 6, 8, 10, 12, 14, 2, 4, 6, 8, 10, 12, 14, 16, 18, 22, …}
Let primorial P(n) = A002110(n) and let r < P(n) be a number such that gcd(r, P(n)) = 1. Thus, r is a residue in the reduced residue system (RRS) of P(n), and the number of r pertaining to P(n) is given by φ(P(n)) = A005867(n). We take the union of the first differences of the r in the RRS of P(n) to arrive at row n of this sequence.
The run lengths of numbers m in the cototient of a number k is tantamount to the first differences between r < k such that gcd(r, k) = 1. The cototient includes any m such that at least 1 prime p | m also divides k, in other words, any m such that gcd(m, k) > 1.
Row n of this sequence is the union of first differences of row n of A286941.
Let L be a primitive first difference as defined above. L is necessarily even since P(n) (for n > 0) is even and all r are odd.
Length of row n = A329815(n).
Table begins:
n: Row
1: (null)
2: 4;
3: 2, 4, 6;
4: 2, 4, 6, 8, 10;
5: 2, 4, 6, 8, 10, 12, 14;
6: 2, 4, 6, 8, 10, 12, 14, 16, 18, 22;
7: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26;
8: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 34;
...
A331119: Indices of A025487(n) in A055932.
1 February 2020 {1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 15, 16, 17, 19, 21, 22, 23, 26, 27, 28, 29, 30, 31, 33, 36, …}.
A055932 lists numbers m whose prime divisors p are consecutive primes starting with 2, while A025487 lists numbers m that are products of primorials. With both, we find a range of indices of primes 1, 2, …, k that divide m. While A055932 admits any multiplicity for primes regardless of their index, the latter only admits decreasing multiplicities as prime index k increases. Therefore, A025487 is a subset of A055932.
A331938: Indices of A002110(n) in A055932.
1 February 2020 {1, 2, 4, 10, 28, 83, 227, 626, 1644, 4290, 11322, 28965, 74469, 189436, 471910, 1166247, 2884920, 7130085, 17349489, 42180190, 101820577, …}.
A055932 lists numbers m whose prime divisors p are consecutive primes starting with 2, admitting multiplicity, while A002110 lists numbers m that are products of distinct consecutive primes starting with 2. Therefore, A002110 is a subset of A055932.
Let 0 ≤ i ≤ k, integers. We can write an efficient algorithm to construct a complete list of all terms m of A055932 up to A002110(k) using A067255(m). Every term m in the list has ω(m) = A001221(m) ≤ k. Starting with A002110(i), we use A067255 to encode m, i.e., the list of multiplicities e pertaining to the 1st…i-th prime pi, allowing position of the multiplicity e in the list to convey pi. Thus, the first “recipe” for m = A002110(i) = {1, 1, …, 1}, a list of i ones. If this does not exceed the limit A002110(k), then we accept it as a value, then increment the last multiplicity. When we have an invalid recipe, we increment the penultimate multiplicity and reset the last to 1, etc., until we have generated all m ≤ A002110(k). As a measure of efficiency, this algorithm generates 1 ≤ m ≤ A002110(12), 74469 terms, in about 2 seconds including sorting, on a 64-bit Intel Xeon E-2286M (2.40 GHz) processor. This is the same amount of time it takes to test numbers 1…400000 to yield the 575 smallest terms of the same sequence.
Vis-á-vis the primorial (A2110), highly composite (A2182), and superabundant (A4394) numbers, we have these relations:
A055932 ⊇ A025487 ⊇ A2110 or A2182 or A4394.
The three sequences intersect but are not subsets/supersets of one another. It is easy to generate A2110 (product of the k smallest primes) but the other two are indices of record transforms based on τ(n) and σ(n). We can use A025487 to make briefer the generation of A2182 and A4394, since A025487 is very much less dense than the set of natural numbers, but the records transform based on the respective functions τ(n) and σ(n) are still of course required.
A332034: V0252: Indices of A002182(n) in A055932 (i.e., V0012(n) in V0210).
5 February 2020 {1, 2, 3, 4, 6, 9, 12, 13, 15, 21, 26, 30, 36, 49, 53, 63, 72, 86, 114, 134, 149, 175, 194, 212, …}.
A055932 lists numbers m whose prime divisors p are consecutive primes starting with 2, while A002182 is a subset of A025487, the latter lists numbers m that are products of primorials. With both, we find a range of indices of primes 1, 2, ..., k that divide m. While A055932 admits any multiplicity for primes regardless of their index, the latter only admits decreasing multiplicities as prime index k increases. A002182 is a subset of A025487, which is in turn a subset of A055932.
A332035: V0253: Indices of A004394(n) in A055932 (i.e., V0512(n) in V0210).
5 February 2020 {1, 2, 3, 4, 6, 9, 12, 13, 15, 21, 26, 30, 36, 49, 53, 63, 72, 86, 114, 149, 175, 212, 221, 285, 367, …}.
A055932 lists numbers m whose prime divisors p are consecutive primes starting with 2, while A004394 is a subset of A025487, the latter lists numbers m that are products of primorials. With both, we find a range of indices of primes 1, 2, ..., k that divide m. While A055932 admits any multiplicity for primes regardless of their index, the latter only admits decreasing multiplicities as prime index k increases. A004394 is a subset of A025487, which is in turn a subset of A055932.
A332241: V0264: Indices of A224078(n) in A025487 (i.e., V1202(n) in V0211).
7 February 2020 {2, 4, 6, 13, 17, 27, 55, 67, 138, 264, 314, 406, 582, 1046, 1835, 3609, 16371, 75611, 118893, 342363}.
Finite and full, since A224078 is finite with 20 terms.
A331478: Irregular table T(n,k) = n − (s−k + 1)² for 1 ≤ k ≤ s, with s = floor(sqrt(n)).
17 January 2020 {0, 1, 2, 0, 3, 1, 4, 2, 5, 3, 6, 4, 7, 0, 5, 8, 1, 6, 9, 2, 7, 10, 3, 8, 11, 4, 9, 12, 5, 10, 13, 6, …}.
Row n begins with n − floor(sqrt(n)).
Zero appears in row n for n that are perfect squares. Let r = sqrt(n). For perfect square n, there exists a partition of n that consists of a run of r parts that are each r themselves; e.g., for n = 4, we have {2, 2}, for n = 9, we have {3, 3, 3}. It is clear through the Ferrers diagram of these partitions that they are equivalent to their Durfee square, thus n − s² = 0.
Since the partitions of any n contain Durfee squares in the range of 1 ≤ s ≤ floor(sqrt(n)) (with perfect square n also including k = 0), the distinct Durfee square excesses must be the differences n − s² for 1 ≤ s ≤ floor(sqrt(n)).
We borrow the term "square excess" from A053186(n), which is simply the difference n − floor(sqrt(n)).
Row n of this sequence contains distinct Durfee square excesses among all integer partitions of n (see example below).
Row n contains distinct terms in row n of A331553.
For n = 4, the partitions are {4}, {3, 1}, {2, 2}, {2, 1, 1}, {1, 1, 1, 1}. The partition {2, 2} has Durfee square s = 2; for all partitions except {2, 2}, we have Durfee square with s = 1. Therefore we have two unique solutions to n − s² for n = 4, i.e., {0, 3}, so row 4 contains these values.
Table begins:
1 0;
2 1;
3 2;
4 0, 3;
5 1, 4;
6 2, 5;
7 3, 6;
8 4, 7;
9 0, 5, 8;
10 1, 6, 9;
11 2, 7, 10;
12 3, 8, 11;
13 4, 9, 12;
14 5, 10, 13;
15 6, 11, 14;
16 0, 7, 12, 15;
17 1, 8, 13, 16;
...
A333238: Irregular table where row n lists the distinct smallest primes p of prime partitions of n.
12 March 2020, offset 2, {2, 3, 2, 2, 5, 2, 3, 2, 7, 2, 3, 2, 3, 2, 3, 5, 2, 3, 11, 2, 3, 5, 2, 3, 13, 2, 3, 7, 2, 3, 5, 2, 3, 5, …}
See expanded study for more.
A prime partition of n is an integer partition wherein all parts are prime. For instance, (3 + 2) is a prime partition of the sum 5; for n = 5, (5) is also a prime partition. For 6, we have two prime partitions (3 + 3) and (2 + 2 + 2). We note that there are no prime partitions for n = 1, therefore the offset of this sequence is 2. The number of prime partitions of n is shown by A000607(n).
For prime n, row n includes n itself as the largest term.
The least primes among the prime partitions of 5 are 2 and 5, cf. the 2 prime partitions of 5: (5) and (3, 2), thus row 5 lists {2, 5}.
The least primes among the prime partitions of 6 are 2 and 3, cf. the two prime partitions of 6, (3, 3), and (2, 2, 2), thus row 6 lists {2, 3}.
Row 7 contains {2, 7} because there are 3 prime partitions of 7: (7), (5, 2), (3, 2, 2). Note that 2 is the smallest part of the latter two partitions, thus only 2 and 7 are distinct.
The plot (x, y) of the primes in row n, with y = n and x = π(p) has an interesting appearance.
Table plotting prime p in row n at pi(p) place, interposing primes missing from row n are shown by "." as a place holder:
n: Row
2: 2
3: . 3
4: 2
5: 2 . 5
6: 2 3
7: 2 . . 7
8: 2 3
9: 2 3
10: 2 3 5
11: 2 3 . . 11
12: 2 3 5
13: 2 3 . . . 13
14: 2 3 . 7
15: 2 3 5
16: 2 3 5
17: 2 3 5 . . . 17
18: 2 3 5 7
19: 2 3 5 . . . . 19
...
Chart rotated 90° counterclockwise, for 2 ≤ n ≤ 240, with rows n in A330507 (Click here for larger, annotated version, or a similar chart with red bars annotated, but larger scope):
A333259: Consider the k primes pj for 1 ≤ j ≤ k in row n of A333238: a(n) = Sum of 2(π(pj) − 1).
16 March 2020, offset 2, {1, 2, 1, 5, 3, 9, 3, 3, 7, 19, 7, 35, 11, 7, 7, 71, 15, 135, 15, 15, 23, 263, …}
In other words, convert the indices of primes pi in row n of A333238 to 1s in the (i − 1)-th place to create a binary number m; convert m to decimal.
The number of prime partitions of n is shown by A000607(n), which in terms of this sequence equates to the number of 1s in a(n), written in binary.
For prime p, row p of A333238 includes p itself as the largest term, since p is the sum of (p); here we find a(p) ≥ 2(p − 1). More specifically, a(2) = 1, a(p) > 2(p − 1) for p odd.
For n = A330507(m), a(n) = 2m − 1, the smallest n with this value in this sequence.
The least primes among the prime partitions of 5 are 2 and 5, cf. the 2 prime partitions of 5: (5) and (3, 2), thus row 5 lists {2, 5}.
The least primes among the prime partitions of 6 are 2 and 3, cf. the two prime partitions of 6, (3, 3), and (2, 2, 2), thus row 6 lists {2, 3}.
Row 7 contains {2, 7} because there are 3 prime partitions of 7: (7), (5, 2), (3, 2, 2). Note that 2 is the smallest part of the latter two partitions, thus only 2 and 7 are distinct.
This sequence is fodder for the plot (x, y) of the primes in row n, with y = n and x = π(p) shown at A333238.
See plot of 10,000 terms (dataset courtesy of Alois P. Heinz).
A334144: Consider the mapping k → (k − (k/p)), where prime p | k. a(n) = maximum distinct terms at any position j among the various paths to 1.
16 April 2020, {1, 1, 1, 1, 1, 2, 2, 1, 2, 2, 2, 2, 2, 2, 3, 1, 1, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 1, 4, …}
(Co-authors: Antti Karttunen, Peter Kagey)
See expanded study for more.
Consider the recursive mapping M of f(n) = n − n/p across primes p | n, ignoring zero differences. We derive totally ordered lists in which the terms mj > m(j + 1), with 0 ≤ j ≤ ℓ. All lists therefore constitute chains C and have the common length ℓ = A064097(n) (see OEIS A333123 comment 2 and A064097 for proof, beyond the scope of this document).
Since ℓ is the same for all chains C produced by recursive f(n), we might arrange the chains vertically (in columns) with antichains A the rows 0 ≤ j ≤ ℓ. This sequence considers only unique terms in the antichains, producing a poset P, of which we might render a Hasse diagram called by some the “Wichita lattice”. Let width W = A064097(n) be the number of chains C, thus the absolute number of nodes is Wℓ.
This sequence thus is the width w of the largest antichain of the poset P, and the largest term in row n of A334184.
Plots of some of the record-setting n appear below: A334144 measures the “width” of the plots (see study).
A334238: Rows n in A334184 that are not unimodal.
19 April 2020, {57, 63, 171, 258, 266, 294, 301, 329, 342, 343, 354, 361, 377, 378, 379, 381, …}
(Co-authors: Antti Karttunen, Peter Kagey)
Consider the mappings f(k) := k → k − k/p across primes p | k.
a(n) = rank levels of antichains in the poset resulting from taking distinct terms generated by the mapping and preserving the order of their generation.
We deem a series of rank levels, such as those of n = 15, i.e., row 15 of A334184 = [1, 2, 3, 2, 1, 1], as unimodal, as the terms increase to a point, then decrease.
Early terms may suggest that 2i ± 1 appear often in a(n). Given 10000 terms, the only such instances are {63, 513, 2047, 16383} for i = {6, 9, 11, 14}.
a(n) for 1 ≤ n ≤ 710 are bimodal. Are there rows n > 710 in A334184 that increase and decrease more than twice?
Example: n = 57 is the smallest number for which rank levels of antichains is not unimodal, under the poset formed from distinct terms resulting from the mapping f(n) := n -> n - n/p across primes p | n.
Hasse diagram Row 57 of A334184
------------- -----------------
57 1
| \
| \
54 38 2
| \/ \
| /\ \
36 27 19 3
| \ | /
| \| /
24 18 2
/| /|
/ | / |
16 12 9 3
| /| /
|/ |_/
8 6 2
| /|
|/ |
4 3 2
| /
|/
2 1
|
|
1 1
Hasse diagrams of the posets P(n) generated by the mappings f(n) for the four smallest terms of A334238:
A333959: First occurrence of n in A334144.
24 April 2020, {1, 6, 15, 33, 65, 77, 154, 161, 217, 231, 455, 469, 483, 693, 957, 987, 1001, …}
(Co-authors: Antti Karttunen, Peter Kagey, Robert G. Wilson v)
Consider the mappings f(m) := m → m − m/p across primes p | m.
Row m of A334184, read as a triangle T(m, k), lists the number of distinct values that proceed from the mapping after exactly k iterations.
A334144(m) is the largest value in row m of A334184.
The smallest term in this sequence that is not an index of a record in A334144 is a(22) = 2093.
Hasse diagrams of the 3 smallest terms, with brackets around the widest row.
[1] 6 15
/ \ /\
/ \ / \
[4 3] 12 __10
| / | \/ |
| / |_/\ |
2 [8 _6 5]
| | /_|_/
| |// |
1 4 3
| /
|_/
2
|
|
1
A334468: List of distinct values of n + A217287(n).
02 May 2020 {4, 6, 8, 12, 15, 16, 18, 24, 27, 30, 32, 36, 40, 45, 48, 54, 60, 63, 64, …}
See expanded study for more.
This sequence is a list of primitive least m > n whose distinct prime factors p are not a subset of those prime factors p found in the range n..(m − 1), i.e., the smallest A217287(n)-smooth number m > n. These numbers serve as "obstructions" that end or break the chains described at A217287.
The number (a(n) − 1) can be found in at least one row of A217438. In other words, this sequence includes any number T(n, A217287(n)) + 1 where T(n, k) is the irregular triangle described at A217438.
Triangle T(n,k) begins:
1 2 3
2 3
3 4 5
4 5 6 7
5 6 7
6 7
7 8 9 10 11
8 9 10 11
9 10 11
10 11 12 13 14
11 12 13 14 15
12 13 14 15
13 14 15
14 15
15 16 17
...
Plot accentuating the last term in the rows above in red: these are numbers in A334468 less 1.
A334469: Indices of zero or positive first differences in A217287.
02 May 2020 {1, 3, 4, 7, 10, 11, 15, 16, 22, 25, 26, 31, 34, 36, 41, 46, 52, 56, 57, 63, …}
See expanded study for more.
Starting with i, we increment i to build a chain of consecutive numbers such that all distinct prime factors of ensuing numbers i + 1, i + 2, etc., divide at least one previous number in the chain. We store the chains in an irregular triangle T(i,j) described in A217438. This sequence lists rows i such that the last term exceeds that of the previous row.
In the graph shown in A334468 above, the blue pixels appear in rows n in A334469.
A333518: a(n) = A000720(A006530(A334468(n))) (i.e., Indices of the greatest prime factor of A334468(n)).
05 May 2020 {1, 2, 1, 2, 3, 1, 2, 2, 2, 3, 1, 2, 3, 3, 2, 2, 3, 4, 1, 4, 2, 3, 3, 2, 3, 2, 3, 4, 2, 3, …}
See expanded study for more.
Consider A334468, a list of numbers m = n+j such that j > 0 is also the smallest number such that n+j has no prime factor > j for some n and j = A217287(n).
Since prime q always contributes a novel prime divisor (i.e., q itself) to the set of distinct primes that divide at least 1 number i the range n + i (1 ≤ i ≤ j), the numbers m in A334468 are composite, and given the above, m is a product of relatively small prime factors.
Table of indices of the first appearance of prime(i) in this sequence at index j. This index j finds the term m at A334468(j) such that gpf(A334468(j)) = prime(i). We give ω(m) (number of distinct prime divisors of m) and Ω(m) (number of prime divisors of m with multiplicity) as well. This table is complete for 1 ≤ j ≤ 2^22 in A334468.
i p(i) j ω Ω A334468(j)
-----------------------------------
1 2 1 1 2 4
2 3 2 2 2 6
3 5 5 2 2 15
4 7 18 2 3 63
5 11 59 3 4 308
6 13 49 3 4 234
7 17 68 3 3 374
8 19 84 2 3 475
9 23 292 3 5 2392
10 29 401 2 4 3625
11 31 518 3 3 4991
12 37 791 4 4 8547
13 41 614 3 5 6232
14 43 615 3 3 6235
15 47 1153 3 5 13912
16 53 1221 4 4 14946
17 59 2654 3 6 39825
18 61 1220 3 4 14945
19 67 2646 3 6 39664
20 71 4965 4 4 87685
21 73 5499 2 3 99937
22 79 4931 4 6 86900
23 83 5879 4 4 108647
24 89 6994 5 5 135102
25 97 10109 4 6 214758
26 101 10444 4 5 223412
27 103 12322 4 5 274804
28 107 13343 4 5 304094
29 109 14088 3 3 325583
30 113 11256 3 9 245888
31 127 19523 4 6 488696
32 131 23472 5 6 614652
33 137 27295 5 6 741444
34 139 22588 4 6 586024
35 149 32147 4 4 908751
36 151 40572 4 6 1213436
37 157 41323 4 8 1240928
38 163 48489 4 5 1515085
39 167 43924 4 4 1340843
40 173 68859 4 6 2331175
41 179 77441 5 5 2692518
42 181 70471 4 6 2398250
43 191 61084 5 6 2016196
44 193 94726 4 6 3449875
45 197 75033 4 6 2589368
46 199 94725 4 6 3449864
47 211 75034 4 7 2589392
...
A334495: Position of prime(n) in A045572, a(1) = a(3) = 0.
27 August 2020 {0, 2, 0, 3, 5, 6, 7, 8, 10, 12, 13, 15, 17, 18, 19, 22, 24, 25, 27, 29, 30, 32, 34, 36, 39, 41, 42, …}
A045572 contains the positive numbers coprime to 10. Nondivisor primes p (i.e., all primes except p | 10, that is, 2 or 5) belong to one of four residues r (i.e., 1, 3, 7, 9) in reduced residue system mod 10. Therefore all primes aside from 2 and 5 appear in A045572.
On account of this fact, one may use A045572 as a sort of prime sieve. This use is less efficient than searching for primes aside from 2 and 3 amid numbers that are ±1 (mod 6), i.e., in A007310, and slightly more efficient than searching for primes aside from 2 amid the odd numbers, but in line with the common (decimal) base.
For prime pn for n ≠ 1 nor n ≠ 3, a(pn) = 4q + (r + 1)/2 − [r > 5] (Iverson brackets), where q = ⌊pn/10⌋ and r = pn mod 10.
a(1) = a(3) = 0 since 2 and 5 are not in A045572.
a(2) = 2 since A045572 = 3, etc.
A333624: Irregular triangle read by rows: S(n,k) = number of triangles of zeros with side length k in the XOR-triangle with first row generated from the binary expansion of n.
08 May 2020 {0, 1, 1, 0, 1, 2, 2, 0, 1, 0, 0, 1, 1, 1, 2, 1, 3, 2, 1, 3, 1, 1, …}
See expanded study for more.
An XOR-triangle is an inverted 0-1 triangle formed by choosing a top row and having each entry in subsequent rows be the XOR of the two values above it, i.e., A038554(n) applied recursively until we reach a single bit.
Let b(n) = n written in binary and let ℓ(n) = 1 + ⌊log2 n⌋ = A070939(n). Let ≥ be a single iteration of XOR across pairs of bits in b(n). Let T(n) be the XOR triangle initiated by b(n). Thus we may refer to any bit in T(n) by the address S(i, j) with 1 ≤ i ≤ ℓ(n) and 1 ≤ j ≤ ℓ(n) − j + 1.
We detect triangles of zeros, which are "voids" amid surrounding 1's or undefined "space" in the t(n) via run lengths of −1 in S(i, j) - S(i − 1, j) for i > 1, and for i = 1, run lengths of zeros.
A334591(n) = length of row n.
Table begins: Read S(n, k) as A067255(r)
0; 1
1; 2
1; 2
0, 1; 3
2; 4
2; 4
0, 1; 3
0, 0, 1; 5
1, 1; 6
2, 1; 12
3; 8
2, 1; 12
3; 8
1, 1; 6
0, 0, 1; 5
0, 0, 0, 1; 7
1, 0, 1; 10
3, 1; 24
1, 2; 18
1, 2; 18
2, 0, 1; 20
...
A334769: a(n) = numbers m that generate rotationally symmetrical XOR-triangles T(m) that have central triangles of zeros.
10 May 2020 {151, 233, 543, 599, 937, 993, 1379, 1483, 1589, 1693, 2359, 2391, 3753, 3785, 8607, 9559, 15017, …}
See expanded study for more.
An XOR-triangle T(n) is an inverted 0-1 triangle formed by choosing a top row the binary rendition of n and having each entry in subsequent rows be the XOR of the two values above it, i.e., A038554(n) applied recursively until we reach a single bit.
A334556 is the sequence of rotationally symmetrical T(n), i.e., RSTs.
A central zero-triangle (CZT) is a field of contiguous 0-bits in T(n) with side length k in A334770, surrounded on all sides by a layer of 1 bits, and generally j > 1 bits of any parity (j in A334796). Alternatively, these might be referred to as "central bubbles".
For n = 151, we have rotationally symmetrical T(151) as below, replacing 0 with “.” for clarity:
1 . . 1 . 1 1 1
1 . 1 1 1 . .
1 1 . . 1 .
. 1 . 1 1
1 1 1 .
. . 1
. 1
1
Correlation of n, a(n), b(n) = A334770(n), and row n of A333624.
Asterisks mark indices j such that A334771(j) = a(n):
n a(n) b(n) A333624(a(n))
---------------------------
1 151 *2 3, 4
2 233 2 3, 4
3 543 *1 1, 0, 0, 3
4 599 *4 6, 3, 0, 1
5 937 4 6, 3, 0, 1
6 993 1 1, 0, 0, 3
7 1379 2 9, 1, 3
8 1483 2 6, 7
9 1589 2 9, 1, 3
10 1693 2 6, 7
11 2359 *3 9, 6, 1
12 2391 *6 9, 3, 0, 0, 0, 1
...
A334770: a(n) = Side length k of the central triangle of zeros in the XOR-triangle T(n).
10 May 2020 {2, 2, 1, 4, 4, 1, 2, 2, 2, 2, 3, 6, 6, 3, 5, 8, 8, 5, 6, 6, 6, 3, 3, 6, 7, 10, …}
See expanded study and entry A334769 for more.
For n = 599, we have a rotationally symmetrical T(599) with k = 4 and j = 2.
1 . . 1 . 1 . 1 1 1
1 . 1 1 1 1 1 . .
1 1 . . . . 1 .
. 1 . . . 1 1
1 1 . . 1 .
. 1 . 1 1
1 1 1 .
. . 1
. 1
1
Since A334769(4) = 599, a(4) = 4.
A334796: a(n) = (A070939(A334769(n)) − A334770(n))/3.
12 May 2020 {2, 2, 3, 2, 2, 3, 3, 3, 3, 3, 3, 2, 2, 3, 3, 2, 2, 3, 3, 3, 3, 4, 4, …}
See expanded study and entry A334769 for more.
a(n) = Frame width j of the central triangle of zeros in the XOR-triangle T(n).
For RSTs, j > 1, since a solid run of ℓ 1s would give rise to a solid run of (ℓ – 1) zeros in iteration m = 2, and every iteration thereafter consists of zeros. Therefore T(2(ℓ – 1) – 1) is not rotationally symmetrical except when ℓ = 1 and T(1) consists of a single 1 bit. Generally,
j = (ℓ – k)/3 = (1 + ⌊log2 n⌋ – k)/3.
Thus the largest possible central zero-triangle is ℓ – 6, or in terms of n, ⌊log2 n⌋ – 5.
Smallest n with j = {2, 3, 4, 5, 6}: {151, 543, 25525, 91534499, 6087484527}, respectively.
First position n of m in a(n), where k = A334770(n):
m n k A334769(n)
-----------------------------
2 1 2 151
3 3 1 543
4 22 3 25525
5 91 12 91534499
6 227 18 60874845273
...
A334771: a(n) = smallest m that generates a rotationally symmetrical XOR-triangle T(m) with a central triangle of zeros with side length n.
10 May 2020 {543, 151, 2359, 599, 8607, 2391, 37687, 9559, 137631, 38231, 602935, 152919, 2202015, …}
See expanded study and entry A334769 for more.
a(n) = Frame width j of the central triangle of zeros in the XOR-triangle T(n).
Correlation of a(n) to indices of a(n) in A334769(i), A334556(j), and row a(n) in A333624:
n a(n) i j A333624(a(n))
--------------------------------------
1 543 3 12 1, 0, 0, 3
2 151 1 10 3, 4
3 2359 11 24 9, 6, 1
4 599 4 13 6, 3, 0, 1
5 8607 15 48 3, 3, 0, 3, 1
6 2391 12 25 9, 3, 0, 0, 0, 1
7 37687 25 76 12, 9, 0, 0, 0, 0, 1
8 9559 16 49 12, 3, 0, 0, 0, 0, 0, 1
9 137631 33 105 6, 6, 0, 3, 0, 0, 0, 0, 1
10 38231 26 77 15, 3, 0, 0, 0, 0, 0, 0, 0, 1
11 602935 49 203 15, 12, 0, 0, 0, 0, 0, 0, 0, 0, 1
12 152919 34 109 18, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1
...
A334836: a(n) = A334769(k) where k is the first position of n in A334796.
13 May 2020 {151, 543, 10707, 33151, 345283, 2213663, 33629695, 134297599, 1109207903, 8657682303, 73283989519, …}. (Offset 2).
This sequence indexes the smallest number m = A334769(k) that, when expressed in binary b(k), generates a rotationally symmetrical XOR-triangle (RST) that features a central zero-triangle (CZT) with frame width n.
A “frame width” is the number of iterations j required to generate the first run of zeros in a CZT of an RST.
Let ℓ = 1 + ⌊log2 n⌋ = A070939(m) for m in A334769. For RSTs, j > 1, since a solid run of ℓ 1s given a recursive XOR function applied to each pair of adjacent bits, would give rise to a solid run of (ℓ − 1) zeros in the next iteration, and every iteration thereafter consists of zeros. Therefore m = (2(ℓ − 1) − 1) is not rotationally symmetrical except when ℓ = 1.
Sequence A334556 lists numbers m that produce RSTs; A334769 those RSTs that feature CZTs. Sequence A334796 renders the frame widths j for numbers in A334769.
a(2) = 151; Rotationally symmetrical XOR-triangle generated by 151, replacing 0s with "." for clarity, showing 2 bits to reach the central zero triangle of side length s = 2:
1 . . 1 . 1 1 1
1 . 1 1 1 . .
1 1 . . 1 .
. 1 . 1 1
1 1 1 .
. . 1
. 1
1
a(3) = 543; RST generated by 543, showing 3 bits to reach the CZT of side length s = 1 = A334770(3):
1 . . . . 1 1 1 1 1
1 . . . 1 . . . .
1 . . 1 1 . . .
1 . 1 . 1 . .
1 1 1 1 1 .
. . . . 1
. . . 1
. . 1
. 1
1
A334896: Read terms ε = S(n,k) in A333624 as Product(pkε) for n in A334769.
23 May 2020 {648, 648, 686, 12096, 12096, 686, 192000, 139968, 192000, 139968, 1866240, …}.
Row a(n) of A067255 = row A334769(n) of A333624.
An XOR-triangle t(n) is an inverted 0-1 triangle formed by choosing a top row the binary rendition of n and having each entry in subsequent rows be the XOR of the two values above it, i.e., A038554(n) applied recursively until we reach a single bit.
Let S(n,k) address the terms in the k-th position of row n in A333624.
This sequence encodes S(n,k) via A067255 to succinctly express the number of zero-triangles in A334769(n). To decode a(n) ⇒ A333624(A334769(n)), we use A067255(a(n)).
a(1) = 648, since b(A334769(1)) = b(151) = 10010111, which generates T(151) as shown below, replacing 1 with "@" and 0 with ".":
@ . . @ . @ @ @
@ . @ @ @ . .
@ @ . . @ .
. @ . @ @
@ @ @ .
. . @
. @
@
In this figure we see 3 zero-triangles of side length k = 1, and 4 of side length k = 2, therefore, T(1,1) = 3 and T(1,2) = 4. This becomes 2^3 * 3^4 = 8 * 81 = 648.
Relationship of this sequence to A334556 and A333624:
n A334769(n) a(n) Row n of A333624
--------------------------------------
1 151 648 3, 4
2 233 648 3, 4
3 543 686 1, 0, 0, 3
4 599 12096 6, 3, 0, 1
5 937 12096 6, 3, 0, 1
6 993 686 1, 0, 0, 3
7 1379 192000 9, 1, 3
8 1483 139968 6, 7
9 1589 192000 9, 1, 3
10 1693 139968 6, 7
11 2359 1866240 9, 6, 1
12 2391 179712 9, 3, 0, 0, 0, 1
13 3753 179712 9, 3, 0, 0, 0, 1
14 3785 1866240 9, 6, 1
15 8607 814968 3, 3, 0, 3, 1
16 9559 2101248 12, 3, 0, 0, 0, 0, 0, 1
...
A334930: Numbers that generate rotationally symmetrical XOR-triangles featuring singleton zero bits in a hexagonal arrangement.
16 May 2020 {1, 11, 13, 91, 109, 731, 877, 5851, 7021, 46811, 56173, 374491, 449389, 2995931, 3595117, …}.
Subset of A334556.
No zero appears in the center of the figure, thus a(n) does not intersect A334769.
Numbers m with A070939(m) (mod 3) = 1 involving alternating run lengths of a singleton zero separated by a pair of 1s in the binary expansion, admitting an initial or final singleton 1.
A diagram montage of 2 ≤ m ≤ 33:
A334931: Numbers that generate rotationally symmetrical XOR-triangles with a pattern of zero-triangles of edge length 2, some of which are clipped to result in some singleton zeros at the edges.
16 May 2020 {151, 233, 1483, 1693, 10707, 13029, 644007, 941241, 317049751, 490370281, 3111314891, …}.
Subset of A334769 which is a subset of A334556.
Numbers m in this sequence A070939(m) (mod 3) = 2. The numbers in this sequence can be constructed using run lengths of bits.
2n has the reverse run length pattern as 2n − 1. a(1) has the run lengths {1, 2, 1, 1, 3}, while a(2) has {3, 1, 1, 2, 1}, etc.
For n = 1 (mod 8): 12..(1132)..113;
For n = 3 (mod 8): 113..(2113)..2112;
For n = 5 (mod 8): 11123..(1123)..1122;
For n = 7 (mod 8): 123112..(3112)..31123, where the parenthetic run lengths occur, when they occur, in multiples of 3. Thus, a(9) has the run length form 12113211321132113 = binary 10010111001011100101110010111 = decimal 317049751.
A diagram montage of 1 ≤ m ≤ 12:
A334932: Numbers that generate rotationally symmetrical XOR-triangles with a pattern of zero-triangles of edge length 3, some of which are clipped to result in some zero-triangles of edge length 2 at the edges.
16 May 2020 {2535, 3705, 162279, 237177, 10385895, 15179385, 664697319, 971480697, 42540628455, …}.
Subset of A334769 which is a subset of A334556.
Numbers m in this sequence A070939(m) (mod 3) = 0. All m have first and last bits = 1.
The numbers in this sequence can be constructed using run lengths of bits thus: 12..(42)..3 or the reverse 3..(24)..21, with at least one copy of the pair of parenthetic numbers.
Thus, the smallest number m has run lengths {1, 2, 4, 2, 3}, which is the binary 100111100111 = decimal 2535.
2n has the reverse run length pattern as 2n − 1. a(3) has the run lengths {1, 2, 4, 2, 4, 2, 3}, while a(4) has {3, 2, 4, 2, 4, 2, 1}, etc.
A diagram montage of 1 ≤ m ≤ 10:
A335077: a(n) sets a record for side length k of zero-triangle in a rotationally symmetrical XOR-triangle.
24 May 2020 {1, 11, 39, 543, 2391, 9559, 38231, 152919, 611671, 2446679, 9786711, 39146839, 156587351, …}
An XOR-triangle T(m) is an inverted 0-1 triangle formed by choosing a top row the binary rendition b(m) of m and having each entry in subsequent rows be the XOR of the two values above it, i.e., A038554(n) applied recursively until we reach a single bit. We may plot T(m) as an equilateral triangle, since each iteration decrements the binary integer length ℓ = 1 + ⌊log2 n⌋ = A070939(m) of the output until we have ℓ = 1.
The XOR function used here requires two inputs; if the inputs agree, the output is 0, else 1.
A rotationally-symmetrical XOR-triangle (RST) is one whose appearance is the same when rotated 120°.
A zero triangle of side length k arises when we have a run of (k + 1) 1s in the preceding iteration.
This sequence contains m that produce T(m) with a recordsetting side length of its largest zero-triangle. For 1 < n < 3, T(a(n)) only has eccentric zero triangles. T(a(4)) has a singleton zero at center, thus a central zero triangle (CZT) of k = 1. For n > 4, all T(a(n)) have CZTs.
The number 543 = A281482(4); we observe that A281482(2i) produces RSTs, and only for 0 ≤ i ≤ 2 do we have eccentric zero triangles larger than any possible CZT. For A281482(2³) = 131583, the side length of its eccentric zero triangles prove much smaller than the largest possible CZT.
Since this sequence wants to maximize the side length k of the largest triangle, we see that the largest triangle possible is the CZT. Let j be the “frame width” or number of iterations required to generate the first run of 0s in the CZT. We note j ≥ 2, since j = 1 requires a run of (k + 1) ones delimited by at least 1 zero; such a width implies that these zeros occur at the beginning and end of b(m). However, beginning binary notation with a leading zero is not permitted. Therefore, if it is possible, we will see T(m) with j > 1.
The numbers that produce record-setting m are the smaller of the binary reverse of m, therefore binary weight favors the least significant digits. Thus we see a 1 followed by a number of zeros in a "prefix" A that, along with a suffix C, must have the same number of bits.
For RSTs with a CZT, we have only one way to produce a solid run of (k + 1) zeros, that is, by dithering bits, which necessitates paired 0 and 1, therefore, we have even k for n > 4.
A run-length formula for a(n) with n > 4 is 12..i(11)..3, meaning that we have 1 one, 2 zeros, any number i of paired 1-0 bits, and a run of 3 ones. Aside from the reversal of this pattern, which puts a greater binary weight in the most significant 3 bits, there is no other way to construct a smaller (or any) CZT with frame size j = 2.
This equates to linear recurrence kernel (5, −4) starting with {2391, 9559} (though 39 is part of the same trajectory).
A333337: Indices of rows of n consecutive smallest primes in A333238, or −1 if n consecutive smallest primes do not appear in A333238.
25 May 2020 {0, 1, 2, 4, 6, 8, 9, 10, 12, 15, 16, 18, 20, 21, 25, 27, 24, 28, 33, 35, 30, 39, 44, 45, 49, 51, …}
Consider the irregular table where row m lists the distinct smallest primes p of prime partitions of m. Row n of this sequence contains all m that have n consecutive primes starting with 2.
Alternatively, positions of k-repunits in A333259.
A330507(n) = First terms in row n.
Null rows occur at n = {90, 151, 349, 352, 444, ...} and are thus filled with the term −1.
Table begins:
0: 0 1
1: 2 4
2: 6 8 9
3: 10 12 15 16
4: 18 20 21 25 27
5: 24 28 33 35
6: 30 39 44 45 49
7: 51 55 63
8: 57 65
9: 60 76 87 91 95
10: 69 75 77 81 85
11: 99 105
12: 111 115 117 119 121
13: 123 125 135
14: 143 145
15: 147 153 155 161 169
16: 159 165 171 175
17: 177 185 187
...
Consider the table plotting prime p in row m of A333238 at pi(p) place, interposing primes missing from row m are shown by "." as a place holder:
m Primes in row m of A333238
---------------------------------
2: 2
3: . 3
4: 2
5: 2 . 5
6: 2 3
7: 2 . . 7
8: 2 3
9: 2 3
10: 2 3 5
11: 2 3 . . 11
12: 2 3 5
13: 2 3 . . . 13
14: 2 3 . 7
15: 2 3 5
16: 2 3 5
17: 2 3 5 . . . 17
...
There are no primes in rows 0 or 1 of A333238, thus row 0 of this sequence contains {0, 1}.
The smallest prime, 2, appears alone in rows 2 and 4 of A333238, thus row 1 of this sequence contains {2, 4}.
We have the primes {2, 3} and no other primes in rows {6, 8, 9} in A333238, thus row 2 of this sequence contains {6, 8, 9}'
We have the primes {2, 3, 5} and no other primes in rows {10, 12, 15, 16} in A333238, thus row 3 of this sequence contains {10, 12, 15, 16}, etc.
A336683: Sum of 2k for all residues k found in the Fibonacci sequence mod n. .
4 October 2020 {1, 3, 7, 15, 31, 63, 127, 175, 511, 1023, 1327, 4031, 7471, 16383, 32767, 43951, 127807, …}
Row n of A079002 compactified as a binary number.
We write a “1” in the (k + 1)-th place for a residue found in the Fibonacci sequence F(m) mod n, and a “0” otherwise, so as to create a characteristic function of residues k in F(m) mod n, if read in a little-endian manner, i.e., such that k = 0 comes first. In this way we can produce a large bitmap of these residues for examination.
Examples:
a(1) = 1 by convention.
a(2) = 3 = 20 + 21, since the Fibonacci sequence mod 2 is {0,1,1} repeated, and 0 and 1 appear in the sequence.
a(8) = 175 = 20 + 21 + 22 + 23 + 25 + 27, since the Fibonacci sequence mod 8 is {0,1,1,2,3,5,0,5,5,2,7,1} repeated, and we are missing 4 and 6, leaving the exponents of 2 as shown.
Binary equivalents of first terms:
n a(n) a(n) in binary
--------------------------
1 1 1
2 3 11
3 7 111
4 15 1111
5 31 11111
6 63 111111
7 127 1111111
8 175 10101111
9 511 111111111
10 1023 1111111111
11 1327 10100101111
12 4031 111110111111
13 7471 1110100101111
14 16383 11111111111111
15 32767 111111111111111
16 43951 1010101110101111
...
Bitmap showing the binary equivalents as in the table above, reversed:
The residue k in F(n) < m is in row m of a189768.
The residue k (mod n + k) is present iff (n + k) = F(2m) and k ≥ 0.
The residue k (mod n – k) is present iff (n + k) = F(m) and k ≥ 0.
There are other structures in the plot with varying slopes, and the horizontal striations associated with the density of residues mod n. Furthermore, there are subtle and soft radial slopes from the origin, with conspicuous clusters of residues e.g., at center (n, k) = (81, 42), (175, 89), (194, 145), or (292, 145).
This sequence is posed in response to my paper on A160136.
A338191: a(1) = 1, a(n) is the least m not already in a(n) such that m mod 10 ≡ decimal digital root of a(n − 1).
15 October 2020 {1, 11, 2, 12, 3, 13, 4, 14, 5, 15, 6, 16, 7, 17, 8, 18, 9, 19, 21, 23, 25, 27, 29, 22, 24, 26, 28, 31, 34, 37, 41, 35, …}
Define d(n) as the decimal digital root of n, which is equivalent to n = r (mod 9), replacing the residue r = 0 with 9 in all cases of nonzero n.
m = 0 (mod 10) is prohibited as a consequence, therefore a(n) is not a permutation of the natural numbers, but contains all positive nonzero m indivisible by 10.
We may write the function d(n) instead as “→” for brevity, separating the least novel m from r with a colon. Therefore, a(2) is derived from a(1) = 1 thus: 1 → 1: 11 (see Example).
Having found a(1)…a(81), we may generate a(81k + j) = 90k + a(j), since a(82) = 91 → 1 and the next interval of 90 unused numbers are congruent to 0 < m < 90 (mod 9). By induction we see the sequence is infinite and contains all nonzero m (mod 90) that are indivisible by 10.
Graphing very many terms results in a line-like plot with slope 10/9. Compare the behavior and plot of this sequence to A248025, which applies d(a(n − 1)) to the first digit of m rather than last.
The sequence repeats 8 phases generally related to → mod 90 by decade.
Phase 1 containing a(n) with 1 ≤ n ≤ 18 and involving 1 ≤ m mod 90 ≤ 19, begins as follows: 1 → 1: 11 → 2: 2 → 2: 12 → 3: 3, etc., therefore we have {1, 11, 2, 12, 3, 13, …, 19}, wherein we have each r twice in succession but incrementing r afterward.
Phase 2 containing a(n) with 19 → n → 27 and involving m mod 90 in the 20s, results from 19 → 2, the third request for r = 2, so a(19) = 21. 21 → 3: 23 → 5: 25, etc. thus {21, 23, 25, 27, 29}, then 29 → 2: 22 → 4: 24, etc. thus {22, 24, 26, 28}.
Phase 3 contains a(n) with 28 ≤ n ≤ 36: 28 → 1: 31 → 4: 34 → 7: 37 → 1: 41 → 5: 35 → 8: 38 → 2: 32 → 5: 45 → 9: 39 → 3: 33 → 6: 36. This exhausts m mod 90 in the thirties. Generally, phases 3 | p involve m mod 90 = 10p + c(p + 1), with 0 ≤ c ≤ 1.
Phase 4 contains a(n) with 37 ≤ n ≤ 45 and begins with {41, 45} already used. 36 → 9: 49 → 4: 44 → 8: 48 → 3: 43 → 7: 47 → 2: 42 → 6: 46. This exhausts m mod 90 in the forties.
Phase 5 contains a(n) with 46 ≤ n ≤ 54: 46 → 1: 51 → 6: 56 → 2: 52, etc., thus {51, 56, 52, 57, 53, 58, 54, 59, 55}, exhausting m mod 90 in the fifties.
Phase 6 contains a(n) with 55 ≤ n ≤ 65: 55 → 1: 61 → 7: 67 → 4: 64 → 1: 71 → 8: 68 → 5: 65 → 2: 62 → 8: 78 → 6: 66 → 3: 63 → 9: 69. We have exhausted m mod 90 in the sixties.
Phase 7 contains a(n) with 66 ≤ n ≤ 72, begining with {71, 78} already used. 69 → 6: 76, etc., thus {76, 74, 72, 79, 77, 75, 73}, exhausting m mod 90 in the seventies.
Phase 8 is the last phase, ending with a(81): 73 → 1: 81 → 9: 89, etc., thus {81, 89, 88, ..., 83, 82}.
Therefore we have generated a(1)…a(81) and may express a(n) for n > 81 via a(81k + j) = 90k + a(j).
A338786: V1205: Numbers in A166981 that are neither superior highly composite nor colossally abundant.
9 November 2020 {1, 4, 24, 36, 48, 180, 240, 720, 840, 1260, 1680, 10080, 15120, 25200, 27720, 110880, …}
These are numbers both highly composite and superabundant but neither superior highly composite nor colossally abundant.
This sequence, A224078, A304234, and A304235 are mutually exclusive subsets that comprise A166981.
Superset A166981 has 449 terms; this sequence has 358, A224078 has 20, A304234 has 39, and A304235 has 32.
Click here for a plot of (x, y) = (x, A2110(A1221(m)) ) with A301414(x) and m highly composite or superabundant, accentuating terms in this sequence.
A338944: Rearrangement of primes into complete Cunningham chains of the second kind, sorted by first prime in the chain.
17 November 2020 {2, 3, 5, 7, 13, 11, 17, 19, 37, 73, 23, 29, 31, 61, 41, 43, 47, 53, 59, 67, 71, 79, 157, 313, 83, 89, 97, 193, 101, 103, 107, 109, …}
A338945: Lengths of Cunningham chains of the first kind that are sorted by first prime in the chain.
17 November 2020 {5, 2, 1, 1, 1, 2, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, …}
A338946: Lengths of Cunningham chains of the second kind that are sorted by first prime in the chain.
17 November 2020 {3, 2, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, …}
A338209: a(1) = 1, a(n) is the least m not already in the sequence whose binary expansion ends with the binary expansion of the binary weight of a(n − 1).
16 December 2020 {1, 3, 2, 5, 6, 10, 14, 7, 11, 15, 4, 9, 18, 22, 19, 23, 12, 26, 27, 20, 30, 28, 31, 13, …}.
Define binary weight wt(n) as A000120(n), the number of 1s in the binary expansion of n. Let w = A000120(a(n − 1)) the binary weight of the previous term. In other words, a(n) is the least m not already in the sequence such that m mod 2k = w, where k = ⌊1 + log2 w⌋.
Michael S. Branicky writes that the sequence is a permutation of the integers since n appears at or before index 2n − 1, the first number with binary weight n, and that 24 appears at index 47201.
The numbers m = 2k with 0 ≤ k ≤ 3 appear at indices {1, 3, 11, 222}. The term 16 has not appeared for n ≤ 214 and may not until n approaches 216.
The numbers m = (2k + 1) appear at indices {2, 4, 12, 223, …}. The numbers m = 2k or (2k + 1) require n approximately equal to 2m in order to appear in the sequence.
The numbers m = (2k − 1) with 1 ≤ k ≤ 14 appear at indices {1, 2, 8, 10, 23, 43, 130, 278, 447, 758, 1390, 2525, 4719, 9333}, respectively.
The plot exhibits dendritic streams of residues r (mod 2k). We can identify coordinates (x, y) = (n, a(n)) on the plot where the streams branch.
The branches of the tree in the plot contain m ≡ to r (mod 2k), where r is a term (except the last term) in row (k − 1) of A049773.
Given 214 terms of this sequence, we see 2 or 3 successive invocations of w, otherwise, w appears just once before a different value succeeds it in the next term.
See paper for analysis of this sequence and related sequences involving the binary-weight found as part of the binary expansion of the next term. This sequence is also nicknamed the “word → least” sequence.
A339024: a(1) = 1, a(n) is the least m not already in the sequence whose binary expansion begins with the binary expansion of the binary weight of a(n − 1).
16 December 2020 {1, 2, 3, 4, 5, 8, 6, 9, 10, 11, 7, 12, 16, 13, 14, 15, 17, 18, 19, 24, 20, 21, 25, 26, …}.
See paper for analysis of this sequence and related sequences involving the binary-weight found as part of the binary expansion of the next term. This sequence is also nicknamed the “word→least” sequence.
We define binary weight wt(n) = A000120(n) as the number of 1s in n2, the number n expressed in binary. Let w = wt(a(n − 1)) the binary weight of the previous term, where w2 is w expressed in binary, and let interval I(j) = 2j ≤ n ≤ (2(j+1) − 1).
Michael S. Branicky writes that the sequence is a permutation of the integers since n appears at or before index 2n − 1, the first number with binary weight n.
The plot (n, a(n)) is organized into streaky clouds that pertains to a “family” M(i) ≤ m < M(i+1) whose binary expansion begins with an odd “prefix” m/2v, where v is the 2-adic valuation of m. There are thus 2v numbers in this range.
The numbers in this range accommodate the binary weights wt(a(n − 1)) = w with 1 <= w ≤ ⌈log2 a(n − 1)⌉ such that w2 appears in part or all of the binary expansion of the prefix m/2v, and perhaps an additional bit in m after the prefix.
Small values of w, for instance w = 1, may appear in any family, but large w require the entire prefix and potentially more (if even).
The w that cannot be found in a particular family are found in a different family that has M(i+1) as its least member.
The families M(i) belong in turn to classes according to odd prefixes. Thus, for example, we may find w = 1, 2, 4, and 9 in class 9, since “1”, “10”, “100”, and “1001” can be found in numbers m that begin, “1001…”.
For w in interval I(j), we have values 1 ≤ k ≤ j − 1 distributed binomially.
Permutation of the natural numbers. We can always find w in a number m in family M(i) that pertains to a class C of numbers that in binary start with the binary expansion of an odd number c.
Numbers m that begin with numbers that are formed of left-trimmed bits of c exhaust the numbers in M(i) before moving to M(i+1) in the same class C.
W
hen we have recordsetting odd w, a new class C opens up based on the binary expansion of a larger odd number c.
See paper for analysis of this sequence and related sequences involving the binary-weight found as part of the binary expansion of the next term. This sequence is also nicknamed the “word → most” sequence.
A339607: a(1) = 1, a(n) is the least m not already in the sequence that contains the binary expansion of the binary weight of a(n − 1) anywhere within its own binary expansion.
16 December 2020 {1, 2, 3, 4, 5, 6, 8, 7, 11, 12, 9, 10, 13, 14, 15, 16, 17, 18, 19, 22, 23, 20, 21, 24, …}
Michael S. Branicky writes that the sequence is a permutation of the integers since n appears at or before index 2n − 1, the first number with binary weight n.
See paper for analysis of this sequence and related sequences involving the binary-weight found as part of the binary expansion of the next term. This sequence is also nicknamed the “word → any” sequence.
Log plot of f(n) = A339607(n) − n) for 1 ≤ n ≤ 214, indicating indices of maxima of f(n) in red and local minima of f(n) in blue.
A340014: V1242: Numbers k in A305056 such that k × A002110(j) is in A004394 for some j ≥ 0
29 December 2020 {1, 2, 4, 6, 8, 12, 24, 48, 72, 120, 144, 240, 288, 360, 720, 1440, 2160, 2880, 4320, …}
Let m be a superabundant number. Since m is a product of primorials P, we may identify a greatest primorial divisor P(ω(m)) = A002110(A001221(A004394(n))).
This sequence lists the primitive quotients k = m/P(ω(m)).
Since m is a product of primorials and k is the quotient resulting from division of m by the largest primorial divisor P, this sequence is also a subset of A025487, which in turn is a subset of A055932.
We can plot all m in A004394 at (A002110(j),k), but this sequence does not accommodate all highly composite numbers; it is missing k = {36, 96, 216, 480, ...}. In contrast, k in A301414 can represent all superabundant numbers m, but a(116)=592424239959167616000 is the least k missing. Therefore in order to plot both A002182 and A004394 one must use the union of a(n) and A301414(n). One can ably plot all the terms common to both A002182 and A004394 (i.e., A166981) using k in A301414.
Plot of m in A4394 (red) or A2182 (blue) at (x, y) → (A340014(x), A2110(y)) for coordinates between (1,0)..(225,379), with (1,0) in lower left corner. See this near-field plot or this wider-field plot for greater detail.
A336684: Irregular triangle in which row n lists residues k found in the sequence Lucas(i) mod n.
7 October 2020 {0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 1, 2, 3, 4, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 7, …}
For row n, it is sufficient to take the union of A000032(i) mod n for 0 <= i <= A106291(n − 1), since the Lucas numbers are cyclical mod n.
Row n contains the Lucas number k < n, and k such that (n + k) is a Lucas number.
Row n for n in A224482 is complete, i.e. it contains all residues k (mod n). This includes n that is a perfect power of 3.
A066981(n) = length of row n.
A223487(n) = n − A066981(n) = number of residues missing from row n.
A224482(n) = rows n that have complete residue coverage, i.e., A066981(n) = n and A223487(n) = 0.
Row 1 contains 0 by convention.
Row 2 contains (0, 1) since the Lucas sequence contains both even and odd numbers.
Row 5 contains (1, 2, 3, 4) since the Lucas numbers mod 5 is {2,1,3,4,2,1} repeated; we are missing the residue 0.
Table begins as shown below, with residue k shown arranged in columns.
n k (mod n)
--------------
1: 0
2: 0 1
3: 0 1 2
4: 0 1 2 3
5: 1 2 3 4
6: 0 1 2 3 4 5
7: 0 1 2 3 4 5 6
8: 1 2 3 4 5 7
9: 0 1 2 3 4 5 6 7 8
10: 1 2 3 4 6 7 8 9
11: 0 1 2 3 4 7 10
12: 1 2 3 4 5 6 7 8 10 11
13: 1 2 3 4 5 6 7 8 9 10 11 12
14: 0 1 2 3 4 5 6 7 8 9 10 11 12 13
15: 1 2 3 4 7 11 14
16: 1 2 3 4 5 7 9 11 12 13 15
A336685: Sum of 2k for residue k in among Lucas numbers mod n.
7 October 2020 {1, 3, 7, 15, 30, 63, 127, 190, 511, 990, 1183, 3582, 8190, 16383, 18590, 47806, …}
Row n of A336684 compactified as a binary number.
a(n) contains even numbers whereas A336683 (pertaining to the Fibonacci sequence) is strictly odd, since 0 is a Fibonacci number but not a Lucas number.
(3 j) = 2^(3 j + 1) − 1 for all j.
A066981(n) = binary weight of a(n).
A223487(n) = n − A066981(n) = number of zeros in the binary expansion of a(n).
a(m) = 2(m + 1) − 1 for m = A224482(n).
a(1) = 1 by convention.
a(2) = 3 = 20 + 21, since the Lucas sequence contains both even and odd numbers.
a(5) = 30 = 21 + 22 + 23 + 24, since the Lucas numbers mod 5 is {2,1,3,4,2,1} repeated, and we are missing 0, leaving the exponents of 2 as shown.
Binary equivalents of first terms:
n a(n) a(n) in binary
--------------------------
1 1 1
2 3 11
3 7 111
4 15 1111
5 30 11110
6 63 111111
7 127 1111111
8 190 10111110
9 511 111111111
10 990 1111011110
11 1183 10010011111
12 3582 110111111110
13 8190 1111111111110
14 16383 11111111111111
15 18590 100100010011110
16 47806 1011101010111110
...
A347285: Irregular triangle T(n,k) starting with n followed by ek = ⌊(logp_k(p(k−1)e(k−1))⌋ such that ek > 0.
27 August 2021 {0, 1, 2, 1, 3, 1, 4, 2, 1, 5, 3, 2, 1, 6, 3, 2, 1, 7, 4, 2, 1, 8, 5, 3, 2, 1, 9, 5, 3, 2, 1, 10, …}
Irregular triangle T(n,k) starting with n followed by ek corresponding to the largest 1 < pkek < p(k−1)e(k−1).
T(0,1) = 0 by convention; 0 is not allowed for n > 0.
T(n,1) = n. T(n,k) is the exponent of pk such that pkT(n,k) is the largest power that does not exceed p(k−1)T(n,k−1).
T(n,k) > T(n,k+1), hence the least first difference among row n of T is 1.
Conjecture: let S be the sum of the absolute values of the first differences of terms in row n. For all n > 0, n − S = 1.
A347284(n) = Product of the pkek in row n.
A089576(n) = row lengths.
A000217(A347286(n)) ≤ sum of terms in row n.
Row 0 contains {0} by convention.
Row 1 contains {1} since we can find no nonzero exponent e such that 3e < 2¹.
Row 2 contains {2,1} since 3¹ < 2² yet 3² > 2². (We assume hereinafter that the powers listed are the largest possible smaller than the immediately previous term.)
Row 3 contains {3,1} since 2³ > 3¹.
Row 4 contains {4,2,1} since 24 > 3² > 5¹, etc.
Triangle begins:
0
1
2 1
3 1
4 2 1
5 3 2 1
6 3 2 1
7 4 2 1
8 5 3 2 1
9 5 3 2 1
10 6 4 3 2 1
11 6 4 3 2 1
12 7 4 3 2 1
13 8 5 4 3 2 1
14 8 5 4 3 2 1
15 9 6 4 3 2 1
16 10 6 4 3 2 1
...
A347286: Row lengths of A347285. Recycled: this sequence is equivalent to A089576
27 August 2021 {0, 1, 2, 2, 3, 4, 4, 4, 5, 5, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 10, 10, 10, 11, 11, 11, …}
a(n) = ω(A347284(n)).
A6939(a(n)) is the largest Chernoff divisor of A347284(n).
A2110(a(n)) is the largest primorial divisor of A347284(n).
A347287: a(n) = Sum_{k = 1..m} 2(ek−1) with 0 < ek = ⌊(logp_k(p(k−1)e(k−1))⌋.
31 August 2021 {1, 3, 5, 11, 23, 39, 75, 151, 279, 559, 1071, 2127, 4255, 8351, 16687, 33327, 66095, 132191, 263263, 526511, 1052847, 2101423, 4202847, …}
Binary compactification of A347285.
a(n) = row sum of 2(m−1) where m are terms in row n of A347285.
a(1) = 1 since we can find no nonzero exponent e such that 3e < 2¹; 2(1−1) = 20 = 1.
a(2) = 3 since 3¹ < 2² yet 3² > 2². (We assume hereinafter that the powers listed are the largest possible smaller than the immediately previous term.) 2(2−1) + 2(1−1) = 2¹+20 = 2+1 = 3.
a(3) = 5 since 2³ > 3¹, hence 2(3−1) + 2(1−1) = 2² + 20 = 4+1 = 5.
a(4) = 11 since 24 > 3² > 5¹, thus 2(4−1) + 2(2−1) + 2(1−1) = 8+2+1 = 11, etc.
n Row n of A347285 a(n)
---------------------------------------------------------------
1: 1 -> 1
2: 1 2 -> 3
3: 1 3 -> 5
4: 1 2 4 -> 11
5: 1 2 3 5 -> 23
6: 1 2 3 6 -> 39
7: 1 2 4 7 -> 75
8: 1 2 3 5 8 -> 151
9: 1 2 3 5 9 -> 279
10: 1 2 3 4 6 10 -> 559
11: 1 2 3 4 6 11 -> 1071
12: 1 2 3 4 7 12 -> 2127
13: 1 2 3 4 5 8 13 -> 4255
14: 1 2 3 4 5 8 14 -> 8351
15: 1 2 3 4 6 9 15 -> 16687
16: 1 2 3 4 6 10 16 -> 33327
...
A347284: a(n) = Product_{j=1..ℓ} pjej with 0 < ej < ⌊log(p(j−1)/log(pj))⌋.
27 August 2021 {1, 2, 12, 24, 720, 151200, 302400, 1814400, 4191264000, 8382528000, 251727315840000, 503454631680000, 3020727790080000, 1542111744113740800000, …}
a(n) is a subset of A025487 which is a subset of A055932. All terms are products of primorials. No primes pj for 1 ≤ j ≤ ℓ have e = 0 with the exception of a(0) = 20.
Let ℓ = ω(a(n)) = A347286(n).
The largest primorial divisor P(ℓ) = A2110(ℓ).
For n > 0, all terms are even.
The greatest prime divisor pℓ has multiplicity eℓ = 1.
All multiplicities e are distinct; for 1 ≤ j ≤ ℓ, the multiplicity ej ≥ ℓ − j + 1.
a(k) | a(n) for 0 ≤ k ≤ n.
The numbers q = a(n+1)/a(n) are primorials.
Finite intersection of A2182 and a(n) = {1, 2, 12, 360, 75600}.
Chernoff number A6939(L) | a(n). Quotient K = a(n) | A6939(ℓ) is in A025487.
The prime shape of terms resembles a simplified map of the US state of Idaho.
a(n) = Product_{j=1..k} pjT(n, j) where T = A347285 and k = A347286(n).
Row n of A347285 yields row a(n) of A067255.
a(0) = 20 = 1;
a(1) = 2¹ = 2, since 3¹ > 2¹
a(2) = 2² × 3¹, since 3¹ < 2² but 3² > 2², and since 5¹ > 3¹;
a(3) = 2³ × 3¹, since 3¹ < 2³ but 3² > 2³, and 5¹ > 3¹;
a(4) = 24 × 3² × 5¹, since 3² < 24 yet 3³ > 24, 5¹ < 3² yet 5² > 3², and 7¹ > 5¹;
etc.
Prime shapes of a(n) for 2 ≤ n ≤ 4:
5 o
4 o 4 x
3 o 3 x 3 x x
2 x 2 x 2 x x 2 x x x
a(2) 1 X X a(3) 1 X X a(4) 1 X X X a(5) 1 X X X X
2 3 2 3 2 3 5 2 3 5 7
This demonstrates that a(n) is in A025487, that A2110(ω(a(n))) is the greatest primorial divisor of a(n) as a consequence (prime divisors represented by capital X’s), and Chernoff A6939(ω(a(n))) | n, prime divisors represented by x’s of any case.
a(n) = A6939(ω(a(n))) * k, k in A025487, represented by o’s.
Because each multiplicity e is necessarily distinct, we may compactify a(n) using Sum_{k=1..ω(a(n))} 2(e−1).
Prime shapes of a(12):
12 o
11 o
10 o
9 o
8 o
7 o o
6 x o
5 x x
4 x x x
3 x x x x
2 x x x x x
a(12) 1 X X X X X X
2 3 5 7 ...
a(12) = A006939(6) × 26 × 3²
= 5244319080000 × 64 × 9
= 3020727790080000.
O
O x
O x x
O x x o x x
O x x o x x o x x x
O x o x x x x o x x x o x x x x
a(1)×6 = a(2)×2 = a(3)×30 = a(4)×210 = a(5)×2 = a(6), etc.
Hence a(n) can be generated by a list of indices of primorials {1, 2, 1, 3, 4, 1, 1, 5, ...} and thereby be efficiently compactified.
A347288: Irregular triangle T(n,k) starting with 2n followed by pkek with 0 < ek = ⌊(logp_k(p(k−1)e(k−1))⌋.
31 August 2021 {1, 2, 4, 3, 8, 3, 16, 9, 5, 32, 27, 25, 7, 64, 27, 25, 7, 128, 81, 25, 7, 256, 243, 125, 49, 11, 512, 243, 125, 49, 11, 1024, 729, 625, 343, 121, 13, 2048, …}
Row n of this sequence lists prime power divisors of A347284(n) from greatest to least.
Row n of this sequence lists prime powers pkek corresponding to ek in row n of A347285.
T(0,1) = 1 by convention.
T(n,1) = 2n. T(n,k) = pkek such that pkT(n,k) is the largest 1 <pkek < p(k−1)e(k−1).
Row 0 contains {1} by convention.
Row 1 contains {2} since we can find no nonzero exponent e such that 3^e < 2^1.
Row 2 contains {4,3} since 3¹ < 2² yet 3² > 2². (We assume hereinafter that the powers listed are the largest possible smaller than the immediately previous term.)
Row 3 contains {8,3} since 2³ > 3¹.
Row 4 contains {16,9,5} since 24 > 3^² > 5¹, etc.
Triangle begins:
2 3 5 7 11 13 17
--------------------------------------------------
0: 1
1: 2
2: 4 3
3: 8 3
4: 16 9 5
5: 32 27 25 7
6: 64 27 25 7
7: 128 81 25 7
8: 256 243 125 49 11
9: 512 243 125 49 11
10: 1024 729 625 343 121 13
11: 2048 729 625 343 121 13
12: 4096 2187 625 343 121 13
13: 8192 6561 3125 2401 1331 169 17
14: 16384 6561 3125 2401 1331 169 17
...
A347354: a(n) = Sum of T(n,k) − T(n-1,k) for irregular triangle A347285.
31 August 2021 {1, 2, 1, 3, 4, 1, 2, 5, 1, 6, 1, 2, 7, 1, 3, 2, 1, 8, 1, 4, 2, 1, 9, 10, 1, 2, 11, 1, 3, 1, 2, 12, 1, 4, 13, 1, 2, 1, 14, 15, 1, 2, 3, 1, 6, 2, 1, 4, 1, 16, 2, 1, 17, 18, 1, 2, 1, 3, 5, 1, 2, 19, 1, 4, 2, 1, 20, …}
Also the length ℓ of d = T(n,k) − T(n-1,k) in row n of A347285 such that d > 1.
Primorial compactification of A347284.
We can construct row n of A347285 by summing a constant array of a(k) 1s for 1 ≤ k ≤ n−1.
A347284(n) = Product_{k=1..n−1} of A2110(a(n)).
Relation of a(n) and irregular triangle A347285, placing "." after the term in the current row where T(n,k) no longer exceeds T(n−1,k). Since the rows of A347285 reach a fixed point of 0, we interpret T(n,k) for vacant T(n−1,k) as exceeding same.
0
1. 1
2 1. 2
3. 1 1
4 2 1. 3
5 3 2 1. 4
6. 3 2 1 1
7 4. 2 1 2
8 5 3 2 1. 5
9. 5 3 2 1 1
10 6 4 3 2 1. 6
11. 6 4 3 2 1 1
12 7. 4 3 2 1 2
13 8 5 4 3 2 1. 7
14. 8 5 4 3 2 1 1
15 9 6. 4 3 2 1 3
16 10. 6 4 3 2 1 2
...
A347355: Least index of n in A347354.
31 August 2021 {1, 2, 4, 5, 8, 10, 13, 18, 23, 24, 27, 32, 35, 39, 40, 50, 53, 54, 62, 67, 70, 73, 85, 89, 94, 99, 100, 104, 105, 115, 129, …}
List of k such that A347286(k) = A347286(k−1) + 1.
Relation of A347354 and irregular triangle A347285, placing "." after the term in the current row where T(n,k) no longer exceeds T(n−1,k). Since the rows of A347285 reach a fixed point of 0, we interpret T(n,k) for vacant T(n−1,k) as exceeding same. The indices n that are highlighted with parentheses are the terms in this sequence.
0: 0
(1): 1. 1
(2): 2 1. 2
3: 3. 1 1
(4): 4 2 1. 3
(5): 5 3 2 1. 4
6: 6. 3 2 1 1
7: 7 4. 2 1 2
(8): 8 5 3 2 1. 5
9: 9. 5 3 2 1 1
(10):10 6 4 3 2 1. 6
11: 11. 6 4 3 2 1 1
12: 12 7. 4 3 2 1 2
(13):13 8 5 4 3 2 1. 7
14: 14. 8 5 4 3 2 1 1
15: 15 9 6. 4 3 2 1 3
16: 16 10. 6 4 3 2 1 2
...
A347356: a(n) = m/A6939(ω(m)) with m = A347284(n).
31 August 2021 {1, 1, 2, 2, 2, 4, 24, 24, 48, 48, 96, 576, 576, 1152, 34560, 207360, 414720, 414720, 829440, 174182400, 1045094400, 2090188800, 2090188800, 2090188800, …}
Quotient of m/A6939(ω(m)), where m = A347284(n), A6939(k) is the k-th Chernoff number, and ω(m) = A1221(m), the number of distinct prime divisors of m.
Primorial index compactification of A347284.
a(n) = A347284(n)/A6939(A347286(n)).
Number of residues r such that prime(n) = p divides A330170(n) for n ≡ r (mod p − 1).
1 August 2020 {1, 1, 2, 1, 2, 1, 1, 2, 4, 4, 1, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 1, 1, 8, 4, 1, 2, 1, 2, 2, …}
Consider A330170(n) = b(n), a sequence that is also a linear recurrence. In that sequence, for every prime p, there exists n ≥ 1 such that p | b(n). Further, for every prime p, p | b(n) for n ≡ −1 (mod p − 1), as a consequence of Fermat's Little Theorem. b(n) is even for all n, 3 | b(n) for even n, and 5 | b(n) for odd n.
Given the examples below, we observe that though p | a(n) for n ≡ −1 (mod p − 1), for certain primes we have additional residues r (mod p − 1) that have p | a(r).
Examples:
Let b(k) = A330170(k).
Generally, for prime p > 3, p | b(k) for k = −1 (mod p − 1).
a(1) = 1 since prime(1) | b(k) for all k, i.e., b(k) is always even.
a(2) = 1 since prime(2) | b(k) for k = 0 mod(prime(2) − 1), i.e., 3 | b(k) for even k. We observe that 3 divides 48 = 24 × 3, 1392 = 24 × 3 × 29, 47448 = 2³ × 3² × 659, etc.
a(3) = 2 since prime(3) | b(k) for k congruent to 1 or 3 mod(prime(3) − 1), i.e., 5 | b(k) for k congruent to 1 or 3 (mod 4), which is tantamount to 5 | b(k) for odd k. We observe that 5 divides 10 = 2 × 5, 250 = 2 × 5³, 8050 = 2 × 5² × 7 × 23, etc.
a(4) = 1 since 7 | b(k) for k = −1 (mod 6). 7 | 8050 = 2 × 5² × 7 × 23.
a(5) = 2 since 11 | b(k) for k congruent to 8 or 9 (mod 10).
…
a(9) = 4 since 23 | b(k) for k congruent to {5, 10, 16, 21} (mod 22).
…
a(25) = 8 since for 97, there are 8 residues {12, 25, 36, 47, 60, 73, 84, 95} (mod 96) for which 97 | b(k).
A347755: Least k that does not appear in A347113(m), 1 ≤ m ≤ n.
12 September 2021 {1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 7, 7, 7, 7, 7, 7, 7, 7, 7, 11, 11, 11, 11, 11, 11, 11, 11, 17, 17, 17, 17, 17, 17, 17, 17, 17, 19, 19, …}
a(0) = 1 by definition, since A347113 = 1 by definition of that sequence.
Lower bound of A347113.
Conjecture: all terms are in A008578. This is true for n ≤ 327680. Let j = A347113(m−1) and k = A347113(m) for k in A347757. For m > 0, k | j.
A347756: Local minima in A347113.
12 September 2021 {1, 2, 3, 7, 11, 17, 19, 23, 31, 37, 59, 61, 67, 79, 97, 151, 157, 199, 211, 229, 271, 307, 337, 367, 499, 577, 601, 619, 691, …}
a(0) = 1 by definition, since A347113 = 1 by definition of that sequence.
Distinct terms in A347755.
Conjecture: subset of A008578.
A347757: Indices of local minima in A347113.
12 September 2021 {1, 8, 11, 20, 28, 37, 51, 53, 101, 116, 136, 146, 159, 213, 302, 318, 440, 520, 638, 698, 702, 912, 1031, 1128, 1528, 1758, …}
a(n)−1 = last instance of A347756(n) in A347755.
a(n+1) > a(n) + 1, since terms in A347113 are distinct by definition.
Sequences that have been entered since 12 September:
A348473: a(n) = Sum_{k=1..A003056(n)} 2(T(n,k)−1), where T(n,k) = k-th term in row n of A235791.
19 October 2021 {1, 2, 5, 9, 18, 35, 69, 133, 266, 523, 1043, 2069, 4133, 8230, 16459, 32843, 65675, 131221, 262421, 524566, 1049127, …}
Decimal value of binary compactification of A235791.
A348475: a(n) = Product_{k=1..A003056(n)} prime(T(n,k)), where T(n,k) = k-th term in row n of A235791.
19 October 2021 {2, 3, 10, 14, 33, 78, 170, 190, 483, 1218, 2046, 4070, 5330, 8385, 33558, 37842, 47082, 127490, 169510, 269445, 825630, 1250886, 1404858, …}
Prime product compactification of A235791. All terms are squarefree.
A348624: a(n) = sum of row n of A348433 expressed as an irregular triangle. .
25 October 2021 {31, 21, 2555, 2805, 3315, 17391, 38893, 104857575, 59363, 2097120, 31713, 376809, 117440484, 18790481885, 197132241, 2885681109, …}
The binary expansion w of a(n) has an interesting appearance shown by the bitmap in links. We may divide w with length m into 3 parts: the most significant part includes all bits including the last 0 before the middle of the word, m/2, a central run of k 1's that includes all but the last 1 before a 0, and a least significant part that includes the last 1 in the central run of 1s and an assortment of 0's. For example, a(3) = 2555 → 100.11111.1011, which we may partition as shown by "." so as to preserve the otherwise-leading 0 in the last part. The central run of 1s generally increases in length as n increases.
A348642: a(n) = Product_{k=1..A003056(n)} prime(T(n,k)), where T(n,k) = k-th term in row n of A235791.
29 October 2021 {2, 4, 12, 24, 72, 240, 720, 1440, 7200, 20160, 60480, 201600, 604800, 1693440, 13305600, 26611200, 79833600, 372556800, 1117670400, …}
Compactification of row n of A237591 via product of prime powers. Row n of A237591 is interpreted instead as row n of A067255, returning index n from that sequence.
All terms are even.
Subset of A055932, but not a subset of A025487, since row n = 14 of A237591 is {8,3,1,2}. It is the least n such that at least one pair of terms in the row exhibit increase.
Intersection with A002182 = {2, 4, 12, 24, 240, 720, 20160} and is finite on account of the prime shape of a(n).
A349191: a(n) = A000720(A348907(n+1)).
9 November 2021 {1, 2, 1, 3, 2, 4, 1, 3, 2, 5, 4, 6, 1, 3, 2, 7, 5, 8, 4, 6, 1, 9, 3, 2, 7, 5, 8, 10, 4, 11, 6, 1, 9, 3, 2, 12, 7, 5, 8, 13, 10, 14, 4, 11, 6, 15, …}
Regarding this sequence as an irregular triangle T(m,j) where the rows m terminate with 1 exhibits row length A338237(m). In such rows m, we have a permutation of the range of natural numbers 1..A338237(m). Records are the natural numbers.
A349192: Irregular triangle T(m,k) = inverse permutation of S(m,k) = A349191 read as an irregular triangle.
9 November 2021 {1, 2, 1, 4, 2, 1, 3, 6, 2, 1, 4, 3, 5, 8, 2, 1, 6, 4, 7, 3, 5, 11, 3, 2, 8, 5, 10, 4, 6, 1, 7, 9, 15, 3, 2, 11, 6, 13, 5, 7, 1, 9, 12, …}
We find k at S(m,k) where S is A349191 read as an irregular triangle. Alternatively, we find prime(k) at U(m,k) where U is A348907 read as an irregular triangle.
A349297: V0250: Triangle T(n,k) = [(2 | n ∧ 2 | k) ∨ (3 | n ∧ 3 | k)], Iverson brackets.
(The “Quincunx” Sequence)
13 November 2021 {0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, …}
Excludes S(n,k) such that gcd(n,k) = 1.
Row n in {1,5} mod 6 consists of k zeros; column k in {1,5} mod 6 is always 0.
Row or column p > 5 where p is prime consists of p zeros.
For n = 0 (mod 6), k in A047229 have T(n,k) = 1.
For k = 0 (mod 6), n in A047229 have T(n,k) = 1.
T(n,k) such that n and k both belong to {2,3,4} mod 6 form a “quincunx” or x-shaped checkerboard pattern evident in the table. In A054521, these have the value 0 along with other terms T(n,k) such that gcd(n,k) > 1.
Table T(n,k) for 1 <= n <= 16, replacing 0 with "." and 1 with "*", showing terms in row n of this sequence.
1: .
2: . *
3: . . *
4: . * . *
5: . . . . .
6: . * * * . *
7: . . . . . . .
8: . * . * . * . *
9: . . * . . * . . *
10: . * . * . * . * . *
11: . . . . . . . . . . .
12: . * * * . * . * * * . *
13: . . . . . . . . . . . . .
14: . * . * . * . * . * . * . *
15: . . * . . * . . * . . * . . *
16: . * . * . * . * . * . * . * . *
---------------------------------------------------
n\k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
 |  |
| The “quincunx” pattern | The pattern in aggregate |
A349298: Positions k in row n of triangles S(n,k) = T(n,k) = 0, where A054521 = S and A349297 = T, or 0 if there are no such k.
9 November 2021 {0, 0, 0, 0, 5, 0, 7, 0, 0, 5, 11, 0, 13, 7, 5, 10, 0, 17, 0, 19, 5, 15, 7, 14, 11, 23, 0, 5, 10, 15, 20, 25, 13, 0, 7, 21, 29, 5, 25, 31, 0, 11, 22, …}
Row n is a list of k for which [A349297 ⊽ A054521] = 1 (Iverson brackets).
Row p > 3 for p prime has the single term p.
Nonzero terms in this sequence are of the form k × m > 1, where 3-smooth k > 1 in A003586 and 5-rough m > 1 in A007310, with m mod 6 ≡ ±1.
Table T(n,k) for 1 <= n <= 16, replacing 0 with "." and 1 with "*", showing terms in row n of this sequence. Rows with no terms are replaced by 0:
1: .
2: . *
3: . . *
4: . * . *
5: . . . . 5
6: . * * * . *
7: . . . . . . 7
8: . * . * . * . *
9: . . * . . * . . *
10: . * . * 5 * . * . *
11: . . . . . . . . . . 11
12: . * * * . * . * * * . *
13: . . . . . . . . . . . . 13
14: . * . * . * 7 * . * . * . *
15: . . * . 5 * . . * 10 . * . . *
16: . * . * . * . * . * . * . * . *
---------------------------------------------------
n\k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Hence, row 5 = {5}, row 7 = {7}, row 11 = {11}, row 13 = {13}, row 14 = {7}, row 15 = {5, 10}, and all other rows 1 <= n <= 16 have no terms, thus are assigned 0 by definition.
Red pixels in this enlarged image indicate the numbers k that appear in row n of this sequence.
A349317: Triangle T(n,k) = [gcd(n, k) > 1], Iverson brackets.
14 November 2021 {0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, …}
Characteristic function of k in the cototient of n, i.e., of A169582: a(A169582(n)) = 1; a(A169581(n)) = 0.
Row sums are in A051953 = n − A000010(n).
A067392 = Sum of k × T(n,k).
A066570 = Product of k in row n such that T(n,k)=1.
Inverse of A054521 = S(n,k); T(n,k) = 1 − S(n,k)
Table T(n,k) for 1 <= n <= 16, replacing 0 with "." for clarity:
1: .
2: . 1
3: . . 1
4: . 1 . 1
5: . . . . 1
6: . 1 1 1 . 1
7: . . . . . . 1
8: . 1 . 1 . 1 . 1
9: . . 1 . . 1 . . 1
10: . 1 . 1 1 1 . 1 . 1
11: . . . . . . . . . . 1
12: . 1 1 1 . 1 . 1 1 1 . 1
13: . . . . . . . . . . . . 1
14: . 1 . 1 . 1 1 1 . 1 . 1 . 1
15: . . 1 . 1 1 . . 1 1 . 1 . . 1
16: . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1
---------------------------------------------------
n\k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
A349405: a(n) = A347113(A347313(n))+1.
16 November 2021{95, 6, 15, 39, 14, 22, 119, 87, 57, 46, 123, 215, 159, 94, 93, 219, 74, 118, 122, 303, 142, 134, 327, 166, 695, 178, 395, 206, 214, 226, …}
These numbers generate primes in A347113.
Let s = A347113, j = s(i)+1, and k = s(i+1). For prime k, j is a squarefree semiprime pq, p < q.
The first 3 primes in s have k = p, while all others observed for i ≤ 219 have k = q.
A349406: a(n) = A349405(n)/A348779(n).
16 November 2021 {19, 3, 5, 3, 2, 2, 7, 3, 3, 2, 3, 5, 3, 2, 3, 3, 2, 2, 2, 3, 2, 2, 3, 2, 5, 2, 5, 2, 2, 2, 3, 7, 2, 2, 3, 3, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 11, …}
Ratio of progenitor and prime in A347113.
Let s = A347113, j = s(i)+1, and k = s(i+1). For prime k, j is a squarefree semiprime pq, p < q.
The first 3 primes in s have k = p, while all others observed for i ≤ 219 have k = q. This sequence thus lists the other prime factor r of j such that r × k = j.
The quasi-linear striations k < n are arranged according to this sequence (see color-coded log-log scatterplot showing primes in green).
A349411: a(n) = prime j = A347113(i) − 1 in order of appearance.
16 November 2021 {2, 5, 11, 23, 47, 3, 7, 13, 19, 17, 31, 37, 29, 59, 41, 83, 167, 43, 61, 53, 107, 67, 71, 73, 79, 89, 179, 359, 719, 1439, 2879, 97, 101, 103, 109, 113, 227, 131, 263, …}
Let s = A347113, j = s(i)+1 and k = s(i+1). We recall the 3 constraints presented in A347113:
1. j = k is forbidden.
2. gcd(j,k) = 1 is forbidden.
3. All terms in s are distinct.
These constraints confine prime j to the relationship j | k, since gcd(j,k)=1 and j=k is forbidden. In the context of s, j | k implies j < k and sequence increase. The least k > j such that j | k is 2j, giving rise to Cunningham chains of the first kind.
A349412: Positions of prime j = A347113(i)+1.
16 November 2021 {1, 2, 3, 4, 5, 8, 9, 13, 17, 21, 25, 31, 34, 35, 39, 40, 41, 45, 56, 60, 61, 66, 70, 75, 79, 88, 89, 90, 91, 92, 93, 102, 105, 108, 113, 121, 122, 126, 127, 133, 140, …}
Let s = A347113, j = s(i)+1 and k = s(i+1). We recall the 3 constraints presented in A347113:
1. j = k is forbidden.
2. gcd(j,k) = 1 is forbidden.
3. All terms in s are distinct.
These constraints confine prime j to the relationship j | k, since gcd(j,k)=1 and j=k is forbidden. In the context of s, j | k implies j < k and sequence increase. The least k > j such that j | k is 2j, giving rise to Cunningham chains of the first kind. The chains are evident in this sequence as series of consecutive positions in s.
A349543: a(n) = A001414(A277272(n)).
21 November 2021 {2, 4, 6, 3, 6, 9, 9, 9, 15, 5, 5, 10, 8, 8, 8, 10, 10, 10, 14, 7, 7, 7, 21, 9, 12, 10, 16, 12, 15, 25, 20, 14, 12, 16, 22, 11, 11, 11, 11, 11, 11, …}
Although terms k in A277272 are distinct, terms m in this sequence may appear A000607(m) times, even consecutively.
The restriction of the number of appearances of m to A000607(m) is a consequence of distinct k such that A001414(k) = m. Distinct k for which A001414(k) = m relates to the number of prime partitions of m and are listed in row m of A064364. For example, k in {7, 10, 12} have A001414(k) = 7. Once these k have appeared in A277272, there is no other way to obtain m = 7 in this sequence. Hence m = 7 is exhausted in this sequence.
Terms are greater than 1.
A348470: Least prime factor of the EKG sequence = A020639(A064413(n)).
6 December 2021 {1, 2, 2, 2, 3, 3, 2, 2, 2, 5, 3, 2, 2, 7, 3, 2, 2, 2, 2, 11, 3, 3, 2, 5, 5, 2, 2, 13, 3, …}
Prime pn appears first at a(A064955(n)). Records are A008578.
A349681: a(1)=1, a(2)=2; for n > 2, a(n) is the least unused positive number k such that j ≠ k and either j | k or k | j, but not both, where j = a(n−1) + 1.
25 November 2021 {1, 2, 6, 14, 3, 8, 18, 38, 13, 7, 4, 10, 22, 46, 94, 5, 12, 26, 9, 20, 42, 86, 29, 15, 32, 11, 24, 50, 17, 36, 74, 25, 52, 106, 214, …}
Variant of A347113.
A350014: V1121: Numbers whose square has a number of divisors coprime to 6.
a(n) = {m : (τ(m²), 6) = 1}.
17 January 2022 {1, 4, 8, 9, 25, 27, 32, 36, 49, 64, 72, 100, 108, 121, 125, 169, 196, 200, 216, 225, 243, 256, 288, 289, 343, 361, 392, 441, 484, 500, 512, 529, 576, 675, 676, 729, 800, 841, 864, …}
a(n) = m in A1694 such that τ(m²) is not divisible by 3, where τ(n) = A5(n).
A051676 ⊂ A350014. ⊂ A1694.
A350741: Records in A095258.
23 January 2022 {1, 3, 4, 6, 9, 27, 34, 68, 94, 235, 289, 578, 799, 1921, 9683, 16021, 27421, 54842, 69301, 138602, …}
A350928: {Partial sums of A095258} + 2
23 January 2022 {3, 6, 8, 12, 18, 27, 54, 72, 80, 85, 102, 136, 204, 216, 240, 250, 275, 286, 299, 322, 329, 376, 470, 705, 720, 736, 768, 816, 867, 1156, …}
Let A095258(n) = m | a(n) such that m is minimal and distinct in A095258.
A350929: a(n) = A350928(n)/A095258(n).
23 January 2022 {3, 2, 4, 3, 3, 3, 2, 4, 10, 17, 6, 4, 3, 18, 10, 25, 11, 26, 23, 14, 47, 8, 5, 3, 48, 46, 24, 17, 17, 4, 3, 18, 52, 73, 27, 10, 74, 112, …}
Log-log scatterplot of A350741(1..n) in red, A095258(1..n) in blue, and A350928(1..n) in gold, for n = 100000.
A347860: Irregular triangle T(n,k) where row n is the partition of n with the least number of 3-smooth parts such that the product of parts is minimal.
23 February 2022 {2, 3, 4, 4, 1, 6, 6, 1, 8, 9, 9, 1, 9, 2, 12, 12, 1, 12, 2, 12, 3, 16, 16, 1, 18, 18, 1, 18, 2, 18, 3, 18, 4, 18, 4, 1, 24, 24, 1, 24, 2, …}
Let k = 2a × 3b be a part such that n is the sum of at least one such k. Then we may represent k at (a,b) on a Cartesian grid. In Dimitrov, et al., these k are known as “digits” in a 2-dimensional number base system. Since these partitions have the least number of parts (digits) in order to represent n, in Dimitrov this is called a “canonic form” for n in base (2,3).
These numbers represent an extreme representation where the digits tend to be spread farthest apart in the plot described above.
Additionally, this canonic representation of n is often identical to the greedy representation of n shown by row n in A276380.
Triangle begins:
1;
2;
3;
4;
4, 1; (product smaller than (3,2))
6;
6, 1; (product smaller than (4,3))
8;
9;
9, 1; (product least of {(9,1), (8,2), (6,4)})
9, 2; (product smaller than (8,3))
12;
...
A348599: Irregular triangle T(n,k) where row n is the partition of n with the least number of 3-smooth parts such that the product of parts is maximal.
23 February 2022 {1, 2, 3, 4, 3, 2, 6, 4, 3, 8, 9, 6, 4, 8, 3, 12, 9, 4, 8, 6, 9, 6, 16, 9, 8, 18, 16, 3, 12, 8, 12, 9, 16, 6, 9, 8, 6, 24, 16, 9, 18, 8, 27, 16, 12, …}
Let k = 2a × 3b be a part such that n is the sum of at least one such k. Then we may represent k at (a,b) on a Cartesian grid. In Dimitrov, et al., these k are known as “digits” in a 2-dimensional number base system. Since these partitions have the least number of parts (digits) in order to represent n, in Dimitrov this is called a “canonic form” for n in base (2,3).
These numbers represent an extreme representation where the digits tend to be close together.
Triangle begins:
1;
2;
3;
4;
3, 2; (product larger than (4,1))
6;
4, 3; (product larger than (6,1))
8;
9;
6, 4; (product greatest of {(9,1), (8,2), (6,4)})
8, 3; (product larger than (9,2))
12;
...
A352064: V3900: Irregular triangle T(n,k) where row n lists the positions of n in A275314.
Row n ∋ {m : m = ∏_{1..ℓ} λ(k) ∧ λ(k) = (p − 1) : p prime ∧ ℓ(n) = A280954(n)}.
2 March 2022 {1, 2, 3, 4, 6, 8, 5, 9, 12, 16, 10, 18, 24, 32, 7, 15, 20, 27, 36, 48, 64, 14, 30, 40, 54, 72, 96, 128, 21, 25, 28, 45, 60, 80, 81, 108, 144, 192, 256, …}
A table of Leonhard Euler’s, cf.
Tentamen Novæ Theoriæ Mvsicæ ex Certissimis Harmoniæ Principiis Dilvcide Expositæ, Typographia Academiæ Scientiarvm, St. Petersburg, Russia (1739), 41.
Associated with gradus suavitatis function A275314.
Triangle begins:
1;
2;
3, 4;
6, 8;
5, 9, 12, 16;
10, 18, 24, 32;
7, 15, 20, 27, 36, 48, 64;
14, 30, 40, 54, 72, 96, 128;
21, 25, 28, 45, 60, 80, 81, 108, 144, 192, 256;
42, 50, 56, 90, 120, 160, 162, 216, 288, 384, 512;
...
A352072: V3701: a(n) = least k such that A003586(n) | 12^k.
8 March 2022 {0, 1, 1, 1, 1, 2, 2, 1, 2, 2, 2, 3, 3, 2, 2, 3, 3, 2, 4, 3, 3, 4, 2, 4, 3, 3, 5, 4, 3, 4, 4, 3, 5, …}
Also, number of digits in the duodecimal expansion of terminating unit fractions 1/A3586.
Let “richness” ρn(m) = ε : m | nε ∧ ε ≥ 0 ∧ (m | nε ∧ m | nδ, ε < δ ∀ δ). Define K as the squarefree kernel of n, that is, A7947(n), the product of all distinct primes p | n. We know that m | Kε iff m | nε. Then RK is the infinite list of m | Kε : ε ≥ 0, as well as all m | nε : ε ≥ 0. We note that A3586: V3500: R6 = 6-regular m, i.e., 3-smooth numbers. This function ρn(x) is described by Hardy & Wright (4th-8th ed.) Theorem 136.
Therefore, let M be the mappings of ρn across RK where K ⊠ n. Therefore we have, for instance, Weisstein’s A117920 = { ρ10 ↦ R10 }. Likewise we have M6 = { ρ6 ↦ R6 } in Zumkeller’s A086415.
For n = p prime, the sequence Mp = ℕ0 = A1477. For n = pε, we have Mn = { ⌊ℕ0/ε⌋ }. Hence, for n = p², Mn = { ⌊ℕ0/2⌋ } = A4526, for n = p³, Mn = { ⌊ℕ0/3⌋ } = A2264, for n = p⁴, Mn = { ⌊ℕ0/4⌋ } = A2265, etc. (These sequences do not merit notes in the respective OEIS entries.)
This sequence is M12 = { ρ12 ↦ R6 }.
A352218: V3711: a(n) = least k such that A003592(n) | 20^k.
8 March 2022 {0, 1, 1, 1, 2, 1, 2, 1, 2, 3, 2, 2, 3, 2, 2, 3, 4, 3, 2, 3, 4, 3, 2, 3, 5, 4, 4, 3, 3, 5, 4, 4, 3, 3, 6, …}
Also, number of digits in the vigesimal expansion of terminating unit fractions 1/A3592.
See A352218. This sequence is M20 = { ρ20 ↦ R10 }.
A352219: V3741: a(n) = least k such that A051037(n) | 60^k.
8 March 2022 {0, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 2, 2, 3, 1, 3, 2, 2, 2, 2, 2, 3, 1, 3, 2, 2, 2, 4, 2, 3, 2, 3, 2, 3, 4, …}
Also, number of digits in the sexagesimal expansion of terminating unit fractions 1/A051037.
See A352219. This sequence is M60 = { ρ60 ↦ R30 }.
A352336: Local minima in A109812.
29 March 2022 {1, 2, 3, 5, 6, 7, 11, 13, 15, 22, 23, 27, 28, 29, 30, 31, 43, 46, 47, 55, 61, 63, 87, 91, 93, 94, 95, …}
A352359: Indices of local minima in A109812.
29 March 2022 {1, 2, 4, 6, 9, 16, 20, 26, 28, 30, 39, 41, 44, 51, 54, 76, 80, 85, 104, 109, 112, 162, 165, 175, …}
A352475: V1120: Numbers m such that τ(m) ⊥ 6.
26 March 2022 {16, 64, 81, 625, 729, 1024, 1296, 2401, 4096, 5184, 10000, 11664, 14641, 15625, 28561, 38416, 40000, …}
m = k² : k ∈ A1694: a(n) = A350014(n)².
Amiram Eldar noted: ∑ 1/A352475 = π2/9 and A290 ∩ A336590.
(This A-number was reserved in honor of my wife’s telephone number! It is the first 6 digits of her local number. Hence I thought it should be home to the most special sequence I could think of, and I dedicate it to Laura Ann.)
A352750: a(n) = binary complement of A109812(n−1) ∧ A109812(n); a(1) = 1.
1 April 2022 {1, 0, 1, 0, 4, 2, 0, 5, 9, 0, 4, 1, 2, 0, 17, 24, 0, 6, 10, 0, 16, 8, 4, 2, 8, 16, 2, 0, 48, …}
b(n) = A109812(n) is the least k that does not appear in b(1..n−1) that avoids the ON bits in b(n−1).
a(n) is the decimal value of available bits “unused” in k.
If b(n) = 2m − 1, then a(n) = 0 iff b(n) = 2m.
A352849: Let seed A(1…3) = {1, 3, 5}. Let m = LCM(A(n−2), A(n−1)) ∧ k = A(n). For n > 3, A(n) = { f(m) → k ∈ ℕ : k⊥m ∧ k ∉ A(1…n−1) ∧ x ∈ ℕ : x⊥m ∧ x ∉ A(1…n−1) ∧ k < x }.
14 April 2022 {1, 3, 5, 2, 7, 9, 4, 11, 13, 6, 17, 19, 8, 15, 23, 14, 25, 27, 16, 29, 21, 10, 31, 33, 20, 37, 39, 22, 35, 41, 12, 43, 47, 18, 49, 53, …}
Axiom 1: k lexically earliest, i.e., if k ∉ A(1…n−1) ⋀ x ∉ A(1…n−1), then k < x, when k and x both satisfy the other axioms.
Axiom 2. k ⊥ m, that is, k belongs to the RRS of m.
Let smallest missing number u : u ∈ ℕ ∧ u ∉ A(1…n−1).
Theorem 1: Equality is banned in A.
Proof: Induction on Axiom 1.
Theorem 2: 2 | A(3j+1) for j > 0.
Proof: The seed terms are odd, therefore u₃ = 2 → A(4) = 2. Now 2 | m for A(5…6), and none of these terms may be even per Axiom 2. We introduce odd terms in A, yet u remains even, so whereupon m is no longer even, an even u enters A. Through induction and because numbers are either even or odd, 2 | A(n) : n > 1 ∧ n mod 3 ≡ 1 for n > 1, i.e., 2 | A(3j+1) for j > 0.
Corollary 2.1: u is even.
Corollary 2.2: There is a smallest missing odd v ∈ ℕ ∧ v ∉ A(1…n−1).
Corollary 2.3: A(3k) and A(3k–1) are odd.
Corollary 2.4: If odd prime q | A(n), then q ⊥ A(n±r) for 1 ≤ r ≤ 2, consequence of axioms.
Theorem 3: Even numbers in A are not in order.
Proof: Let p < q be odd primes. Since we have 2 consecutive odd predecessor terms, theorem 1 and axiom 2 delay 2ps : (p, m) > 1 in favor of 2qs : (q, m) = 1.
Theorem 4: Let q be an odd prime. q | A(n) → q ⊥ A(n±r) : 1 ≤ r ≤ 2, but q may or may not divide A(n±3).
Proof: We see q | A(n) → q ⊥ A(n±r) : 1 ≤ r ≤ 2 for the same reason 2 | A(n) → 2 ⊥ A(n±r) : 1 ≤ r ≤ 2. However, since we may rewrite q | A(n) as A(n) mod q ≡ 0 and since generally N mod q is in {0…q−1}, we have a degree of freedom for q that is not available for 2. Furthermore, since 2 | u and q > 2, if there is an even u : q ⊥ u (or, say u < w : p|u ∧ q|w), though q ⊥ m, on account of Axiom 1 we have k = u instead of q | k.
Theorem 5: Let q be an odd prime. In a certain interval a ≤ n ≤ b, we either have 2q | A(n) or q | A(n) but not both.
Proof: Since 2 | A(3j+i) is confined to i = 1 and since q | A(3j+i) may only again appear for j ≥ 4, we have 2q | A(3j+1) or q | A(3j+i)with i ≠ 1. For q = 3, this is especially relevant, since 6 | A(3j+1) determines the appearance of even or odd numbers divisible by 3 in the sequence.
Corollary 5.1: 6 | A(3j+1) → 3 ∤ A(3j+i) : 2 ≤ i ≤ 3 and vice versa.
Open question: does j take another value for q = 3 and k ≥ 85? If not, then this sequence is not a permutation of natural numbers. If so, then perhaps this sequence has episodes of drawing in terms with the factor 6.
A353383: V3800: Duodecimal proper regulars P12.
15 April 2022 {1, 2, 3, 4, 6, 8, 9, 16, 18, 27, 32, 54, 64, 81, 128, 162, 256, 243, 486, 512, 1024, 729, 1458, 2048, 4096, …}
M₁₂ = ρ₁₂ ↦ R₆ ∧ P₁₂ = { k ∈ M₁₂ : 12 ∤ k}.
P₁₂ = { k ∈ M₁₂ : 12 ∤ k}.
Hence row n includes { 22w−1, 22w, 3w, 2 × 3w. }
A353384: V3810: Vigesimal proper regulars P20.
15 April 2022 {1, 2, 4, 5, 10, 8, 16, 25, 50, 32, 64, 125, 250, 128, 256, 625, 1250, 512, 1024, 3125, 6250, 2048, 4096, …}
M20 = ρ20 ↦ R10 ∧ P20 = { k ∈ M20 : 20 ∤ k}.
P20 = { k ∈ M20 : 20 ∤ k}.
Hence row n includes { 22w−1, 22w, 5w, 2 × 5w. }
A353385: V3841: Sexagesimal proper regulars P60.
15 April 2022 {1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 8, 9, 16, 18, 24, 25, 36, 40, 45, 48, 50, 72, 75, 80, 90, 100, 144, 150, 200, 225, 400, 450, …}
M60 = ρ60 ↦ R30 ∧ P60 = { k ∈ M60 : 60 ∤ k}.
P60 = { k ∈ M60 : 60 ∤ k}.
We plot powers 2x on an axis shown here proceeding from asterisk (which indicates the origin, the empty product a.k.a., 1) to the bottom right, 3y on an axis pointing to the bottom left, and powers 5z on an axis pointing up. The volume indicated is infinite, but grows in “shells” that expand 2 units on the 2 axis, and 1 unit each on the 3 and 5 axes. This explains the weird increase in the number of terms per row.
Looking at the z = 0 plane, we have 4 more terms than in the previous row. At the highest z plane, we also have 4 more terms than previously. We can say that the parallelepiped extension involves 4 more terms than before, that happens to be ω(60) × Ω(60) = 4 × 3 = 12 additional terms per level. Thus, we have A017641(w) terms in each row for w > 0.
For row w, plot terms m = 2^x * 3^y * 5^z at (x,y,z). Rows are labeled below the figures parenthetically for clarity. The x axis points toward the bottom right, the y axis to the bottom left, and the z axis upward. In the plot, we mark terms from previous rows by ".", and use "*" to show the origin, that is, the empty product 1:
125
375 250
1125 750 500
3375 2250 1000
6750 2000
25 . 4000
75 50 . . 8000
225 150 100 . . .
450 200 675 . .
400 1350 .
5 . . 800
15 10 . . . . 1600
30 20 45 . . . . .
90 40 135 . .
80 270 .
1 * * * 160
3 2 . . . . 320
6 4 9 . . . . .
12 18 . 8 27 . . .
36 24 16 54 . . .
72 48 108 . . 32
144 216 . 96 64
432 288 192
864 432
1728
(0) (1) (2) (3)
The terms in row w are sorted, hence row 1 has {2, 3, 4, 5, 6, 10, 12, 15, 20, 30}.
A353035: A119435(2n); local minima in A119435.
18 April 2022 {1, 2, 3, 4, 6, 8, 11, 15, 20, 27, 36, 48, 64, 85, 116, 153, 208, 273, 366, 493, 649, 888, 1161, 1579, …}.
A119435(2k) is a local minimum. See Theorem 2. Observe that A030101(2k) = 1. 2k expressed in binary is 1 followed by zeros. When we reverse this number, the leading zeros are trivial and we read the number 1 in the 2⁰ place. Therefore we set A(n) = U(2k, 1), which by definition is the smallest missing number m ∉ A(1…n−1).
Bitmap of A353035(n), n = 1..45, 4× exaggeration. Here we show 1 in black and 0 in white, with the least-significant bit at bottom.
A353036: Maxima in A119435.
18 April 2022 {1, 2, 5, 9, 13, 17, 23, 29, 33, 43, 51, 53, 61, 65, 83, 95, 107, 113, 125, 129, 163, 183, 199, 203, 219, 233, 237, 253, 257, …}.
Terms are odd except for a(2) = 2.
(2(k+1) – 1) ± 2 ∈ A353036. See Corollary 4.1.
Bitmap of A353036(n), n = 1..967, 12× exaggeration. Here we show 1 in black and 0 in white, with the least-significant bit at bottom.
A353037: Maxima in A119435.
18 April 2022 {1, 2, 3, 5, 7, 9, 11, 15, 17, 19, 23, 27, 31, 33, 35, 39, 47, 55, 63, 65, 67, 71, 79, 87, 95, 111, 119, 127, 129, …}.
Terms are odd except for a(2) = 2.
2k±1 ∈ A353037 for k > 0. See Theorem 3.
A119435(2k + 1) =A119435(2k − 1) + 4. See Corollary 3.1.
Bitmap of A353037(n), n = 1..967, 12× exaggeration. Here we show 1 in black and 0 in white, with the least-significant bit at bottom.
A352097: (Y1379): A(1) = 4, A(2) = 9; let i = A(n−2) and j = A(n−1); A(n) = least k not already in the sequence such that (j, k) = 1 and 1 < (i, k) ≠ i ≠ k.
3 June 2022 {4, 9, 10, 21, 8, 15, 14, 25, 6, 35, 16, 45, 22, 27, 20, 33, 26, 51, 28, 39, 32, 57, 34, 63, 38, 49, 12, 77, 18, 55, 24, 65, …}.
Completely neutral restriction of the Yellowstone sequence axioms. Restricted to constitutive states ① (symmetric semicoprimality), ③⑦ (assymetric semicoprimality in semidivisibility, i.e., mixed neutrality), and ⑨ (symmetric semidivisibility).
Theorem: the sequence is in composites. Proof: Prime k must either divide or be coprime to i, but 1 < (i, k) ≠ k precludes k | i. Therefore there is no avenue for primes in the sequence.
All i and k must have a constitutively neutral relationship, that is, the “unrelated” relationship as seen in A045763.
Theorem: if prime p | j then p does not divide k. Consequence of coprimality axiom (j, k) = 1. Hence, even terms are nonadjacent in the sequence. Therefore we begin this sequence with {4, 9}.
A version of Yellowstone sequence S = A098550 that strips out features attributable to prime S(n) and their cototient successors S(n+2). In S, those 2 groups produce quasi-rays in scatterplot that have primes appear late and their successors early in S.
Composite quasi-rays in the Yellowstone sequence scatterplot are retained, bifurcated according to parity for same reasons as in that sequence.
Analogous to A240024 and its relationship to A064413.
Conjecture: permutation of the composite numbers.
A352098: (Y1): A(1) = 6, A(2) = 15; let i = A(n−2) and j = A(n−1); A(n) = least k not already in the sequence such that (j, k) = 1 and 1 < (i, k) = m > 1, and both ω(i) and ω(k) exceed ω(m).
3 June 2022 {6, 15, 14, 33, 10, 21, 22, 35, 12, 55, 26, 45, 28, 39, 20, 51, 38, 63, 34, 57, 40, 69, 44, 75, 46, 65, 18, 85, 52, 95, 24, 115, …}.
Symmetrically semicoprime (constitutive state ①) restriction of the Yellowstone sequence axioms.
Theorem: the sequence contains numbers in A024619. Proof: with m = (i, k) > 1, we have omega(m) ≥ 1, however, both ω(i) and ω(k) must exceed ω(m). Therefore, if no number in the sequence has a single distinct prime factor, none can arise. This restricts the sequence to numbers that are not prime powers.
Theorem: i and k are nondivisors of one another. Proof: both ω(i) and ω(k) exceed ω(m), therefore there exists some prime p | i and some prime q | k, yet, p does not divide k and q does not divide i. Hence i does not divide k and k does not divide i.
The numbers i and k have the relationship described in A272619, a sort of relationship described in A045763.
Analogous to A337687 regarding its relationship to A064413.
Theorem: if prime p | j then p does not divide k. Consequence of coprimality axiom (j, k) = 1. Hence, even terms are nonadjacent in the sequence. Therefore we begin this sequence with {6, 15}.
Composite quasi-rays in the Yellowstone sequence scatterplot are retained, bifurcated according to parity for same reasons as in that sequence.
Conjecture: permutation of A024619.
A353916: (E2347): A(1) = 1, A(2) = 2; let j = A(n−1) and m = ω((j, k)) > 1; A(n) = least k not already in the sequence such that min(ω( j), ω(k)) = m < max(ω( j), ω(k)).
10 May 2022 {1, 2, 6, 3, 12, 4, 10, 5, 15, 9, 18, 8, 14, 7, 21, 27, 24, 16, 20, 25, 30, 32, 22, 11, 33, 66, 36, 42, 28, 49, 35, 70, …}.
Completely neutral restriction of the EKG sequence axioms. Restricted to constitutive states ②④ (asymmetric semicoprimality in divisibility) and ③⑦ (assymetric semicoprimality in semidivisibility, i.e., mixed neutrality). Constitutive opposite of A337687.
See this paper for an extensive study.
Let P = { distinct p | j }, and let Q = { distinct q | k }. Let g = (j, k) > 1 and let G = {P ∩ Q}. Noncoprime j and k implies |G| > 0. This sequence is such that |G| > 0, |P| > |G|, and |Q| = |G|, or vice versa.
Coprimality and equality are forbidden, forcing primes into divisibility. Because of this, A(i) = mp → A(i+1) = p → A(i+2) = sp, with composite m and s not powers of p > 2. For n > i, multiples of p including powers may appear in the sequence. Consequently, odd primes enter the sequence late.
Numbers m that immediately precede and follow prime p have ω(m) > 1.
For any adjacent pair of terms with n > 1, if one is prime then the other cannot be a power of that prime, since such a pair would have the same number of distinct prime divisors.
This sequence requires an asymmetric version of the relation of j and k seen in A337687. In that sequence, we have P != Q, |P| > |G| and |Q| > |G|, therefore we have symmetry in that there is at least 1 prime p | j that does not divide k, and at least 1 prime q | k that does not divide j. That sequence occurs among composites m with ω(m) > 1, but this sequence admits primes, since, for |G| = 1, we must have |P| or |Q| equal to 1, and there is no prohibition for multiplicity to exceed 1. A353917 is a version of this sequence that prohibits divisibility, hence primes do not appear, but composite prime powers do.
Open question: do the primes appear in order? (They do for n ≤ 2¹⁶).
A353917: (E37): A(1) = 1, A(2) = 2; let j = A(n−1) and m = ω((j, k)) > 1; A(n) = least k not already in the sequence such that min(ω( j), ω(k)) = m < max(ω( j), ω(k)), but neither j | k nor k | j.
10 May 2022 {4, 6, 8, 10, 16, 12, 9, 15, 25, 20, 30, 18, 27, 21, 49, 14, 32, 22, 64, 24, 42, 28, 70, 40, 60, 36, 66, 44, 110, 50, 90, 48, 78, 52, 128, 26, 169, …}.
Mixed neutral restriction of the EKG sequence axioms. Restricted to constitutive states ③⑦ (assymetric semicoprimality in semidivisibility, i.e., mixed neutrality).
The sequence exhibits phases involving alternating composite prime powers and squarefree semiprimes. These manifest in log-log scatterplot in a caustic fashion, where the composite prime power is very much larger than the squarefree semiprime for sufficiently large n.
Let P = { distinct p | j }, and let Q = { distinct q | k }. Let g = (j, k) > 1 and let G = {P ∩ Q}. Noncoprime j and k implies |G| > 0. This sequence is such that |G| > 0, |P| > |G|, and |Q| = |G|, or vice versa, yet neither j | k nor k | j.
Theorem: primes are prohibited. Proof: since we have ( j, k) > 1 and do not allow divisibility, and since primes must either divide or be coprime to another number m, primes do not appear in this sequence.
Theorem: squarefree semiprimes j = pq are followed by k = p² or k = q². Proof: since ω(j) = |P| = 2 and is squarefree, we have 2 cases pertaining to successor k, both with (j, k) > 1.
1.) |P| = |G| ⇒ |Q| > |G| ∧ |Q| > |P|.
2.) |Q| = |G| ⇒ |P| > |G| and |P| > |Q|.
The first case implies some prime r | k yet (j, r) = 1. But this would require j | k, which is prohibited. The second case suggests either p | k or q | k, but so as to satisfy non-divisibility axiom, we are forced into either k = pe, e > 1, or k = qm, m > 1.
Corollary: powers of the same prime appear in natural order in this sequence.
There is a weaker alternation between numbers in A120944 and A350352 as n is sufficiently large. This alternation exhibits prime power factor features akin to the composite prime power-squarefree semiprime alternation.
Conjecture: permutation of composite numbers.
Log-log scatterplot of A(1..2¹⁶), showing records in red, local minima in blue, accentuating composite prime powers in green and squarefree semiprimes in gold.
A353954: a(0) = 1; a(n) = A019565(A109812(n)).
12 May 2022 {1, 2, 3, 5, 6, 7, 10, 21, 11, 15, 14, 33, 35, 22, 105, 13, 30, 77, 26, 55, 42, 65, 66, 91, 110, 39, 70, 143, 210, 17, …}
Interpretation of A109812 written in binary instead as if written in "multiplicity notation", that is, as if we write 1 if divisible by prime(k+1), otherwise 0 in the k-th place. Example, decimal 12 is written in binary as 1100 = 2² + 2³, and take exponents 2 and 3 and instead construe them as prime(2+1) × prime(3+1) = 5 × 7 = 35.
Permutation of squarefree numbers A5117.
A353955: a(n) = A019565(A353709(n)).
12 May 2022 {1, 2, 3, 5, 7, 6, 11, 35, 13, 22, 15, 91, 17, 10, 21, 143, 34, 105, 19, 26, 33, 85, 14, 39, 55, 119, 78, 95, 77, 51, 65, 154, 57, 221, 70, …}
Interpretation of A353709 written in binary instead as if written in "multiplicity notation", that is, as if we write 1 if divisible by prime(k+1), otherwise 0 in the k-th place. Example, decimal 12 is written in binary as 1100 = 2² + 2³, and take exponents 2 and 3 and instead construe them as prime(2+1) × prime(3+1) = 5 × 7 = 35.
If A353709 is a permutation of nonnegative numbers, then this sequence is a permutation of squarefree numbers A5117.
A354799: V1131: m ∈ A1694 such that 3 | τ(m²), where τ(n) = A5(n).
21 June 2022 {16, 81, 128, 144, 324, 400, 432, 625, 648, 784, 1024, 1152, 1296, 1936, 2000, 2025, 2187, 2401, 2500, 2592, 2704, 3200, 3456, 3600, 3888, 3969, 4624, 5000, …}
A = { A001694 \ A350014 } = { m ∈ A001694 : τ(m²) mod 3 ≡ 0 }.
Σn≥1 1/a(n) = ζ(2)×ζ(3)/ζ(6) − 5×ζ(3)/(2×ζ(2)) = 0.1166890133… (Dr. Eldar).
A352394: (Y24): a(n) = n for n ≤ 3; let i = a(n−2) and j = a(n−1); a(n+1) = least k not already in the sequence such that (j, k) = 1 and (i, k) = m > 1 and only one of either ω(i) or ω(k) exceed ω(m), and either i | k or k | i.
23 June 2022 {1, 2, 3, 10, 21, 5, 7, 15, 14, 165, 182, 11, 13, 22, 39, 110, 273, 55, 91, 220, 819, 4, 9, 20, 63, 260, 693, 26, 33, 130, 231, 65, 77, 195, 154, 3315, 2926, …}
A restriction on the Yellowstone sequence A098550 analogous to A113552 regarding its relationship to A064413.
Semicoprime-divisor (constitutive states ②④) restriction of the Yellowstone sequence axioms.
Theorem: i | k implies i < k, otherwise k | i implies i > k, a consequence of definition.
Theorem: Prime i implies i < k, since prime i is forced into i | k. Conversely, prime k implies i > k.
Theorem: even terms cannot be adjacent. Proof: If prime p | j, then p cannot divide k as well, because then (j, k) ≥ p and by definition of “prime”, p > 1, which contradicts the axiom (j, k) = 1. Since 2 is prime, consecutive even terms are prohibited.
Conjecture: sequence is not a permutation of natural numbers. Proof sketch: Since either i | k or k | i, and defining m as the smaller of the 2 terms, as n increases, it becomes harder to reach all numbers through multiplication or division by m. Therefore it would seem that there is strong tendency for the sequence to fall into multiplicative recurrence as does A113552.
A354720: (Y2347): a(n) = n for n ≤ 3; let i = a(n−2) and j = a(n−1); a(n+1) = least k not already in the sequence such that (j, k) = 1 and (i, k) = m > 1 and only one of either ω(i) or ω(k) exceed ω(m)
23 June 2022 {1, 2, 3, 10, 21, 4, 7, 6, 35, 8, 5, 12, 55, 9, 11, 15, 22, 25, 16, 45, 14, 27, 32, 33, 20, 81, 64, 39, 28, 13, 42, 65, 18, 125, 66, 85, 24, 17, 30, 119, 36, 49, 60, 77, 40, 121, …}
A restriction on the Yellowstone sequence A098550 analogous to A353916 regarding its relationship to A064413.
Semicoprime-regular (constitutive states ②③④⑦) restriction of the Yellowstone sequence axioms.
Theorem: even terms cannot be adjacent. Proof: If prime p | j, then p cannot divide k as well, because then (j, k) ≥ p and by definition of prime, p > 1, which contradicts the axiom (j, k) = 1. Since 2 is prime, consecutive even terms are prohibited.
A355149: Partial sums of A351743. (with Sigrist and Sycamore)
21 June 2022 {1, 2, 4, 5, 10, 12, 15, 20, 24, 27, 28, 56, 57, 114, 116, 145, 150, 156, 169, 170, 340, 341, 682, 684, 855, 860, 1032, 1035, 1150, 1152, 1161, 1204, 1232, …}
a(n) is odd iff n mod 3 ≡ 1. Parity of a(n) is related to that of A351743(n).
Conjecture (by Sigrist): a(n + 42) = 3645 × a(n) + b(n) for n ≥ 138 (where b is 42-periodic).
A354853: (Y37): a(1) = 4, a(2) = 9; let i = a(n−2) and j = a(n−1); a(n+1) = k such that (j, k) = 1 and (i, k) = m > 1 and only one of either ω(i) or ω(k) exceed ω(m), and neither i | k nor k | i.
23 June 2022 {4, 9, 10, 21, 8, 27, 14, 15, 16, 25, 6, 35, 32, 49, 12, 77, 30, 121, 18, 55, 42, 125, 24, 65, 64, 169, 20, 39, 70, 81, 28, 33, 128, 243, 22, 45, 256, 105, 26, 63, 512, …}
A restriction on the Yellowstone sequence A098550 analogous to A353917 regarding its relationship to A064413. This sequence exhibits phases similar to those in A353917, except between every other term instead of adjacent terms.
Mixed neutral (constitutive states ③⑦) restriction of the Yellowstone sequence axioms.
Let P = {p | i : p prime}, and let Q = {q | k : q prime}. Let g = (i, k) > 1 and let G = {P ∩ Q}. Noncoprime i and k implies |G| > 0. This sequence is such that |G| > 0, |P| > |G|, and |Q| == |G|, or vice versa, yet neither i | k nor k | i.
Theorem: terms are composite. Proof: since divisibility and coprimality between i and k is prohibited and since primes must either divide or be coprime to other numbers, no primes appear in the sequence.
Theorem: even terms cannot be adjacent. Proof: If prime p | j, then p cannot divide k as well, because then (j, k) ≥ p and by definition of prime, p > 1, which contradicts the axiom (j, k) = 1. Since 2 is prime, consecutive even terms are prohibited. Hence we start the sequence with {4, 9}.
Theorem: squarefree semiprimes i = pq are followed by k = p² or k = q². Proof: since ω(i) = |P| = 2 and is squarefree, we have 2 cases pertaining to successor k, both with (i, k) > 1.
1.) |P| == |G| implies |Q| > |G| and |Q| > |P|.
2.) |Q| == |G| implies |P| > |G| and |P| > |Q|.
The first case implies some prime r | k yet (i, r) = 1. But this would require i | k, which is prohibited. The second case suggests either p | k or q | k, but so as to satisfy non-divisibility axiom, we are forced into either k = pε ε > 1, or k = qm, m > 1.
A355058: V1130: Numbers m such that τ(m) mod 6 ≡ 3.
4 July 2022 {4, 9, 25, 36, 49, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 676, 784, 841, 900, 961, 1089, 1156, 1225, 1369, 1444, 1521, 1600, …}
All terms are square; contains squares of primes.
A355058 =
{ A290 \ A352475 }.
A355701: a(n) = Product of prime(k+1) where k runs through the exponents of the positions 2k of the 1-bits in A354169(n)
14 July 2022 {1, 2, 3, 5, 7, 6, 11, 13, 17, 35, 19, 23, 29, 22, 31, 39, 37, 41, 43, 85, 47, 133, 53, 59, 61, 667, 67, 71, 73, 62, 79, 33, 83, 481, 89, 97, 101, 1763, …}
Compactification of A354169. Offset matches A354169.
A356626: Position of A332979(n) in the Doudna sequence A5940.
24 August 2022 {1, 2, 4, 7, 15, 29, 61, 125, 249, 497, 1009, 2033, 4081, 8177, 16369, 32753, 65521, 131057, 262081, 524225, 1048513, 2097089, 4194241, 8388545, 16777153, …}
Offset to match A332979.
Let n₂ be the binary expansion of n with a length of ℓ bits. Let W(n) = A120(n) the binary weight of n, i.e., the number of ones in n₂, while Z(n) = ℓ − W(n) be the number of zeros in n₂. Let Q be the number of runs of ones in n₂, L(k) be the run length of the k-th least significant run of ones, and P(k) the partial sum of the number of zeros to the right of the k-th run of ones.
Define the Doudna function f(n) = ∏_{k=1..Q} prime(P(k)+1)L(k). The Doudna sequence A5940(n) = s(n) = f(n−1) with s(1) = 1.
Theorem 1: the maximum of s(n) for n = 2(k−1)+1..2k is a prime power.
Proof. s(n) corresponds to f(m), m = n−1, hence m = 2(k−1)+1..2k. The number m in this domain has k bits. Binary numbers involve zeros and ones, thus, k = W(m) + Z(m). It is clear that W(m) = Ω(f(m)) and Z(m) = π(GPF(f(m)))−1. Hence we have k = Ω(f(m)) + π(GPF(f(m)))−1. We maximize f(m) for a k-bit number m when the greatest prime factor of f(m) has maximum multiplicity, therefore the maximum of s(n) for n = 2(k−1)+1..2k is a prime power.
Theorem 2: s(2k−2 j+1) = prime(j+1)(k−j), j = 1..k−1.
Proof: We write (2k−2 j)₂ as (k−j) ones followed by j zeros, a k-bit binary number. We have Q=1 run of ones in (2k−2 j)₂. Therefore f(2k−2 j) = prime(P(1)+1)L(1) = prime(j+1)(k−j), that is, row k of A180944.
Hence the maximum of s(n) for n = 2(k−1)+1..2k is tantamount to the maximum of row k of A180944.
A356804: Binary encoded version of A356803.
6 September 2022 {0, 0, 1, 3, 6, 14, 28, 31, 59, 123, 243, 499, 995, 2019, 2028, 2045, 4061, 4095, 8127, 16319, 32575, 65343, 130623, 261695, 523327, 1047615, 2095167, 4192319, …}
Let S(n) = list of forbidden primes for A354790(n); A356803(n) is the product of these primes. Then a(n) = Sum of 2(i−1) over all prime(i) in S(n).
Conversely, if a(n) has binary expansion a(n) = ∑ b(i) × 2i, b(i) = 0 or 1, then S(n) consists of { prime(i+1) : b(i) = 1 }. (After comment by N. J. A. Sloane at A354765)
Analogous to A354765.
A357148: a(n) = A357082(n−1) + A357082(n).
15 September 2022 {1, 3, 5, 7, 9, 15, 16, 15, 16, 24, 29, 32, 33, 29, 34, 29, 32, 36, 34, 42, 61, 64, 34, 32, 61, 64, 61, 64, 61, 64, 65, 72, 76, 64, 72, 85, 76, 64, 72, 82, 64, 72, 100, …}
Auxiliary sequence for A357082.
A357149: a(n) = smallest missing number in A357082(k) for k = 0..n.
15 September 2022 {1, 3, 5, 7, 9, 15, 16, 15, 16, 24, 29, 32, 33, 29, 34, 29, 32, 36, 34, 42, 61, 64, 34, 32, 61, 64, 61, 64, 61, 64, 65, 72, 76, 64, 72, 85, 76, 64, 72, 82, 64, 72, 100, …}
Auxiliary sequence for A357082.
Smallest missing number is indicated by amber line in the chart below. Black terms are those of A357082.
A357150: Primitive terms in A357148.
15 September 2022 {1, 3, 5, 7, 9, 15, 16, 24, 29, 32, 33, 34, 36, 42, 61, 64, 65, 72, 76, 82, 85, 91, 100, 104, 116, 127, 128, 129, 133, 144, 146, 153, 154, 169, 172, 179, 192, 209, …}
Auxiliary sequence for A357082.
A356627: Primes whose powers appear in A332979.
Primes p whose powers are maxima for row n in T(n,k) = prime(k)(n−k+1).
27 September 2022 {2, 3, 5, 7, 11, 17, 29, 37, 41, 59, 67, 71, 97, 127, 149, 191, 223, 269, 307, 347, 419, 431, 557, 563, 569, 587, 593, 599, 641, 727, 809, 937, 967, 1009, 1213, 1277, 1423, …}
Maxima of row n > 0 of A5940, A182944, and A182945 are powers of these primes.
Indices k of primes p = A40(k), listed here show an interesting correlation with the function f(k) = A40(k) − A302334(k). (Peter Munn)
5 | A332979(5..7), thus 5 is in the sequence.
7 | A332979(8), thus 7 is in the sequence.
A357942: a(1) = 1, a(2) = 2. Thereafter, if there are prime divisors p | a(n−1) that are coprime to a(n−2), a(n) is the least novel multiple of the product of these primes. Otherwise a(n) is the least novel multiple of the squarefree kernel of a(n−1). See comments.
22 October 2022 {12, 44, 98, 3174, 844, 22020, 217070, 1092747, 8870024, 262315467, 221167422, 47255689915, 82462576220, 1043460553364, 79180770078548, …}
Let k be the greatest common squarefree divisor of a(n−2) and a(n−1) and let s = A7947(a(n−1)). If k = 1, then a(n) = ms × s, else a(n) = mk × k, where mi is the smallest multiple of i such that m × i ∉ a(1..n−1).
Variant of A357963; a(21) = 36, but A357963(21) = 22.
Variant of EKG sequence A064413.
A356322: V7013: a(n) is the smallest number that starts a run of exactly n consecutive tantus numbers m (i.e., m ∈ A126706), or −1 if no such number exists.
28 October 2022 {1, 2, 4, 6, 3, 9, 12, 8, 10, 5, 15, 18, 14, 7, 21, 24, 16, 20, 25, 30, 36, 42, 28, 56, 70, 35, 105, 27, 33, 11, 22, 26, 13, 39, 45, 40, 32, 34, 17, 51, …}
See this paper for context, and this paper for some background.
Term a(n) begins a run of n consecutive tantus numbers, i.e., nonsquarefree numbers m such that ω(m) > 1.
The run of m must occur between successive primes.
Problem is similar to Catalan’s Conjecture regarding consecutive multus numbers.
There are 4 consecutive tantus numbers m starting from 844 and again from 2888, but since 848 and 2892, respectively, are also in A126706, these m ascribe to n = 5 instead. The range m = 3174..3177 has at most n = 4 numbers in A126706 and 3174 is the smallest number with that quality, hence a(4) = 3174.
A358089: V7012: First differences of tantus numbers A126706.
31 October 2022 {6, 2, 4, 4, 8, 4, 4, 1, 3, 2, 2, 2, 2, 4, 3, 5, 4, 3, 1, 4, 4, 4, 2, 2, 4, 2, 1, 1, 4, 4, 4, 4, 1, 3, 4, 2, 6, 3, 1, 4, 4, 3, 1, 2, 2, 1, 3, 4, 2, 2, 4, …}
See this paper for context.
A356322 relates to the first instances of exactly k consecutive 1s in this sequence.
a(n) − 1 = number of 0s between 1s in A355447.
For prime p, m such that m mod p², unless m = pε, ε > 1, is tantus (i.e., m ∈ A126706), as a consequence of definition of A126706. Therefore m ≤ 4 is common, m ≤ 9 much less so. Consequently, the arrangement of A126706 mod M for M in A061742 presents a quasi-modular pattern as seen in the example and raster link at A355447.
a(51265) = 7; m = 9 is not observed in the first 6577230 terms of the sequence, a dataset corresponding to terms k ≤ 2²⁴ in A126706.
A358173: V8012: First differences of plenus numbers A286708.
1 November 2022 {36, 28, 8, 36, 52, 4, 16, 9, 63, 36, 68, 8, 32, 9, 43, 16, 76, 72, 27, 1, 108, 16, 64, 36, 68, 4, 28, 89, 36, 27, 4, 69, 71, 27, 29, 20, 72, 77, …}
See this paper for context.
Consider the sequence of powerful numbers A1694 ⊃ A246547, the sequence of composite prime powers (multus numbers). Let s = A1694(k) such that ω(s) > 1 be followed by t = A1694(k+1) such that ω(t) = 1.
Since A286708 = A001694 \ A246547, prime powers t are missing in A286708. We consider s = A286708(j) and note that the difference A286708(j+1) − A286708(j) > A1694(k+1) − A001694(k).
Therefore we see a subset S ∋ s in A286708 that plots “out of place” with respect to the complementary subset R = A286708 \ S; some of this subset S (red) exceeds the maxima of R (blue) in the scatterplot of this sequence. The plot of R resembles the scatterplot of A1694.
A358174: Smaller of a pair of numbers (m, m+1) such that both are products P of composite prime powers with ω(P) > 1.
1 November 2022 {675, 9800, 235224, 465124, 1825200, 11309768, 384199200, 592192224, 4931691075, 5425069447, 13051463048, 221322261600, 865363202000, 8192480787000, 11968683934831, …}
See this paper for context.
In other words, smaller of a pair of numbers such that both are plenus numbers.
This question is related to Catalan’s Conjecture (1844) that 8 and 9 are the only adjacent multus numbers. (Catalan’s conjecture was proved by Preda Mihăilescu in 2002). It appears that there are many adjacent plenus numbers.
We can pose a sequence {8} ∪ A358174 comprising the smaller of a pair of numbers that are adjacent powerful numbers m ∈ A1694.
a(n) = A286708(k), where k is a position of 1 in A358173.
a(1) = 675, since 675 = 3³ × 5² and 676 = 2² × 13² are the smallest products of at least 2 composite prime powers that differ by 1. 675 and 676 are adjacent plenus numbers.
a(2) = 9800 since (3⁴ × 11²) − (2³ × 5² × 7²) = 1.
a(3) = 235224 since (5² × 97²) − (2³ × 3⁵ × 11²) = 1.
A358258: First n-bit number to appear in Van Eck’s sequence (A181391).
5 November 2022 {0, 2, 6, 9, 17, 42, 92, 131, 307, 650, 1024, 2238, 4164, 8226, 17384, 33197, 67167, 133549, 269119, 525974, 1055175, 2111641, 4213053, 8444257, 16783217, …}
See this brief.
A358259: Positions of the first n-bit number to appear in Van Eck’s sequence.
5 November 2022 {1, 5, 10, 24, 41, 52, 152, 162, 364, 726, 1150, 2451, 4626, 9847, 18131, 36016, 71709, 143848, 276769, 551730, 1086371, 2158296, 4297353, 8607525, 17159741, …}
See this brief.
A358001: V1124: Numbers whose number of divisors is coprime to 210.
3 December 2022 {1, 1024, 4096, 59049, 65536, 262144, 531441, 4194304, 9765625, 43046721, 60466176, 241864704, 244140625, 268435456, 282475249, 387420489, 544195584, 1073741824, …}
See this brief.
A358250: V1125: Numbers whose square has a number of divisors coprime to 210.
3 December 2022 {1, 32, 64, 243, 256, 512, 729, 2048, 3125, 6561, 7776, 15552, 15625, 16384, 16807, 19683, 23328, 32768, 46656, 62208, 100000, 117649, 124416, 161051, 177147, 186624, 200000, …}
See this brief.
A358786: a(1) = 1. For n > 1, a(n) is least novel k ≠ n such that rad(k) = rad(n) and either k | n or n | k, i.e., k ⑥⑧ n.
8 December 2022 {1, 4, 9, 2, 25, 12, 49, 16, 3, 20, 121, 6, 169, 28, 45, 8, 289, 36, 361, 10, 63, 44, 529, 48, 5, 52, 81, 14, 841, 60, 961, 64, 99, 68, 175, 18, 1369, …}
Variant of A358971 that additionally requires either k | n or n | k. This version eliminates nondivisor n and a(n) seen in a scatterplot of A358971. First differs from A358971 at n = 18.
Some consequences of definition:
There are no fixed points outside of a(1) = 1.
Prime power pε implies a(pε) = p(ε+1) for odd ε, else p(ε−1). Hence a(p) = p² comprise maxima, while a(p²) = p comprise minima.
Let lpf(m) = least prime factor of m. Squarefree m implies a(m) = lpf(m) × m and a(lpf(m) ×m) = m, as seen in scatterplot in rays with slope p and 1/p, respectively. Therefore squarefree numbers are sequestered along or below a(n/2) = n/2.
Let κ = rad(n); a(n) and n (such that a(n) ≠ n) belong to the same sequence κ × Rκ, where Rκ is the list of κ-regular numbers, 1 and those whose prime divisors are restricted to p | κ. For example, if κ = 6, then a(n) and n belong to 6 × A3586, and if κ = 10, then a(n) and n belong to 10 × A3592.
A355432: Let c(n) be the trajectory generated by mapping the arithmetic derivative A3415 until and including the first instance of 0 or a number having the factor of the form pp with prime p. a(n) is the sum of terms in c(n).
5 January 2023: {0, 1, 3, 4, 4, 6, 12, 8, 8, 21, 18, 12, 12, 14, 35, 23, 16, …}
Abandoned due to lost motivation and error, then recycled.
A359243: a(1) = 1, a(2) = 2. Let j = a(n−1). For n > 2, a(n); for n > 2, if j is prime then a(n) = least novel k such that φ(k)/k < φ(j)/j, else a(n) = least novel k such that φ(k)/k > φ(j)/j.
16 January 2023 {1, 2, 6, 3, 4, 5, 8, 7, 9, 11, 10, 13, 12, 14, 15, 17, 16, 19, 18, 20, 21, 23, 22, 25, 29, 24, 26, 27, 31, 28, 32, 33, 35, 37, 30, …}
Permutation of natural numbers.
Once we have a(n) = prime(m), we require a(n+1) = u, the smallest missing number. Thereafter, we find the smallest k new to the sequence such that φ(k)/k > φ(j)/j until we reach another prime. Primes p appear in order, since φ(p)/p = (p−1)/p.
Sequence can be interpreted as an irregular triangle where row 0 = {1} and row m > 1 begins with prime(m). In such a triangle, we observe prime(m) > min(row m) for m > 5, yet we can find prime(m) either less than or exceeding max(row m) for 2b⁰ terms of this sequence, or m = 1..82032.
Since φ(k)/k ascribes to squarefree kernel κ = rad(k) = A7947(k) and κ < mκ where mκ is nonsquarefree yet rad(m) | κ, squarefree k appear before mκ. For example, a(3) = 6 and a(13) = 12; a(11) = 10 and a(20) = 20, etc.
Odd prime numbers tend to appear early, even numbers tend to appear late.
A359382: a(1) = 1, a(2) = 2. Let j = a(n−1). For n > 2, a(n); for n > 2, if j is a primorial, then a(n) = least novel k such that φ(k)/k > φ(j)/j, else a(n) = least novel k such that φ(k)/k < φ(j)/j.
16 January 2023 {1, 2, 6, 3, 4, 10, 12, 30, 5, 8, 14, 18, 42, 60, 210, 7, 9, 15, 16, 20, 24, 66, 84, 90, 330, 420, 2310, …}
Abandoned 15 March until I can put in place a cogent fast code, was only able to get 100 terms using a relatively naive program.
A357910: The natural numbers ordered lexically by their prime factorization, with prime factors written in decreasing order (see comments).
23 January 2023 {1, 2, 4, 3, 6, 8, 9, 12, 5, 10, 15, 30, 16, 27, 18, 25, 20, 45, 60, 7, 14, 21, 42, 35, 70, 105, 210, 32, 81, 24, 125, 40, 75, 90, 49, 28, 63, 84, 175, 140, 315, 420, …}
See this brief.
Project: Sequences having to do with constitutive state counting functions (January-April 2023).
A355432: V5101: Symmetric semidivisor counting function ξ₉(n) = k ⑨ n |, k < n. a(n) = number of k < n such that rad(k) = rad(n) and k does not divide n.
22 February 2023 {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 4, …}
See this brief. This sequence counts the number of k shown in black dots in row n shown in red at right. The numbers n in red are strong tantus numbers in A360768, while the numbers k in blue at bottom are tantus numbers in A126706.
A360768: V0700: “Strong tantus” numbers: numbers k that are neither prime powers nor squarefree, such that k/rad(k) ≥ q, where prime q = A119288(k).
22 February 2023 {18, 24, 36, 48, 50, 54, 72, 75, 80, 90, 96, 98, 100, 108, 112, 120, 126, 135, 144, 147, 150, 160, 162, 168, 180, 189, 192, 196, 198, 200, 216, 224, 225, 234, 240, …}
A360768 ⊂ A126706. Numbers k such that there exists j such that 1 ≤ j ≤ k and rad(j) = rad(k), but j does not divide k.
See this brief.
A360589: V5102: Highly symmetrically semidivisible numbers. Numbers k that set records in A355432.
22 February 2023 {1, 18, 48, 54, 162, 384, 486, 1350, 1458, 2250, 2430, 3750, 6000, 6750, 7290, 11250, 12150, 14580, 15000, 15360, 18750, 21870, 30720, …}
Subset of A055932.
For n > 1, subset of A360768, which is in turn a subset of A126706.
See this brief.
A360767: V0701: “Weak tantus” numbers: numbers k that are neither prime powers nor squarefree, such that k/rad(k) < q, where prime q = A119288(k).
28 February 2023 {12, 20, 28, 40, 44, 45, 52, 56, 60, 63, 68, 76, 84, 88, 92, 99, 104, 116, 117, 124, 132, 136, 140, 148, 152, 153, 156, 164, 171, 172, 175, 176, 184, 188, 204, …}
A360767 ⊂ A126706. Numbers k such that there does not exist j such that 1 ≤ j ≤ k and rad(j) = rad(k), but j does not divide k.
See this brief.
A360769: V7005: Odd tantus numbers.
28 February 2023 {12, 20, 28, 40, 44, 45, 52, 56, 60, 63, 68, 76, 84, 88, 92, 99, 104, 116, 117, 124, 132, 136, 140, 148, 152, 153, 156, 164, 171, 172, 175, 176, 184, 188, 204, …}
(A5408 ∩ A360768) ⊂ A126706.
A360480: V6101: Symmetric semitotative counting function ξ₁(n) = k ① n |, k < n.
28 February 2023 {0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 3, 3, 0, 0, 3, 0, 5, 5, 6, 0, 6, 0, 8, 0, 9, 0, 5, 0, 0, 8, 11, 7, 10, 0, 13, 10, 13, 0, 12, 0, 16, 13, 17, 0, 16, 0, 18, 14, 20, …}
a(n) = number of numbers k < n, with gcd(k, n) > 1, such that there is at least one prime divisor p | k that does not divide n, and at least one prime divisor q | n that does not divide k.
See this brief.
Table listing k ≤ n counted by row n = 10…22 of this sequence:
a(10) = 1: 6 . . . .
a(11) = 0: . . . . . .
a(12) = 1: . . . . 10 . .
a(13) = 0: . . . . . . . .
a(14) = 3: 6 . . . 10 . 12 . .
a(15) = 3: 6 . . . 10 . 12 . . .
a(16) = 0: . . . . . . . . . . .
a(17) = 0: . . . . . . . . . . . .
a(18) = 3: . . . . 10 . . . 14 15 . . .
a(19) = 0: . . . . . . . . . . . . . .
a(20) = 5: 6 . . . . . 12 . 14 15 . . 18 . .
a(21) = 5: 6 . . . . . 12 . 14 15 . . 18 . . .
a(22) = 6: 6 . . . 10 . 12 . 14 . . . 18 . 20 . .
A360765: V0701: “Thick tantus” numbers: numbers k that are neither prime powers nor squarefree, such that A7947(k) × A053669(k) < k.
5 March 2023 {36, 40, 45, 48, 50, 54, 56, 63, 72, 75, 80, 88, 96, 98, 99, 100, 104, 108, 112, 117, 135, 136, 144, 147, 152, 153, 160, 162, 171, 175, 176, 184, 189, 192, 196, 200, 207, 208, 216, …}
Let rad(k) = A7947(k), and let q = A053669(k).
Let j = A7947(k) × A053669(k) = rad(k) × q.
Composite prime powers pε such that ε > 1 and pε > 4 have the property j < k.
With rad(pε) = p, in the case of p = 2, pq = 6, 6 < 2ε for ε > 2. In the case of odd p, we have 2p < pε for ε > 1.
Squarefree k do not have this property, since rad(k) = k, thus, kq > k by definition of prime q.
For k in this sequence, ω(j) > ω(k), but Ω(j) ≤ Ω(k).
Subset of A126706.
See this brief.
A360912: V5103: Records in A355432.
5 March 2023 {0, 1, 2, 4, 8, 10, 14, 16, 21, 23, 26, 33, 34, 39, 44, 51, 52, 54, 55, 58, 67, 70, 76, 77, 80, 83, 84, 95, 98, 104, 119, 124, 133, 134, 142, 148, 153, 160, 164, 168, …}
See this brief.
A360543: V6201: Asymmetric (mixed) semitotative counting function ξ₃(n) = k ③ n |, k < n.
6 March 2023 {0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 6, 0, 0, 0, 0, 11, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 5, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 1, 26, …}
a(n) = number of numbers k < n, gcd(k, n) > 1, such that ω(k) > ω(n) and rad(n) | k.
See this brief.
List of asymmetric semitotatives for n shown at left. This sequence counts the number of these semitotatives.
8: 6
9: 6
16: 6, 10, 12, 14
25: 10, 15, 20
27: 6, 12, 15, 18, 21, 24
32: 6, 10, 12, 14, 18, 20, 22, 24, 26, 28, 30
36: 30
40: 30
45: 30
48: 30, 42
49: 14, 21, 28, 35, 42
50: 30
54: 30, 42
56: 42
63: 42
64: 6, 10, 12, 14, 18, 20, 22, 24, 26, 28, 30, 34, 36,
38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62;
etc.
A361235: V5201: Asymmetric semidivisor counting function ξ₇(n) = k ⑦ n |, k < n. a(n) = number of k < n such that rad(k) = rad(n) and k does not divide n.
6 March 2023 {0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 2, 0, 2, 1, 0, 0, 3, 0, 2, 1, 3, 0, 2, 0, 3, 0, 2, 0, 10, 0, 0, 2, 4, 1, 4, 0, 4, 2, 3, 0, 11, …}
a(n) = number of k < n, such that k does not divide n, ω(k) < ω(n) and rad(k) | rad(n).
a(n) = 0 for prime powers, since the definition implies ω(n) ≥ 2.
See this brief.
List of asymmetric semidivisors for n shown at left. This sequence counts the number of these semidivisors.
6: 4
10: 4, 8
12: 8, 9
14: 4, 8
15: 9
18: 4, 8, 16
20: 8, 16
21: 9
22: 4, 8, 16
24: 9, 16
26: 4, 8, 16
28: 8, 16
30: 4, 8, 9, 12, 16, 18, 20, 24, 25, 27
etc.
A361098: A360765 ∩A360768.
V74: “Panstitutive numbers”: Numbers n such that ∃ j, k : (j ¦¦ n ∧ k ◊¦ n).
15 March 2023 {12, 18, 24, 18, 36, 20, 40, 12, 24, 36, 48, 48, 54, 45, 50, 60, 18, 36, 54, 72, 28, 56, 40, 80, 24, 48, 72, 96, 98, 90, 84, 75, 54, 96, 108, 63, 60, 90, 120, 50, 100, …}
These numbers have both at least 1 symmetric semidivisor (j ¦¦ n, i.e., j ⑨ n) and at least 1 asymmetric (mixed neutral) semitotative (k ◊¦ n, i.e., k ③ n).
Therefore they have all 8 possible constitutive relations between k ≤ n. (⓪①③④⑤⑥⑦⑨)
Hence we call these “pan-stitutive numbers”.
Table pertaining to the first 12 terms:
Key: a = A360480, b = A360543, c = A243823; d = A361235, e = A355432, f = A243822;
g = A046753 = f + c, tau = A5, phi = A10.
n | a + b = c | d + e = f | g + tau + phi - 1 = n
------------------------------------------------------
36 | 10 + 1 = 11 | 4 + 1 = 5 | 16 + 9 + 12 - 1 = 36
48 | 16 + 2 = 18 | 3 + 2 = 5 | 23 + 10 + 16 - 1 = 48
50 | 18 + 1 = 19 | 4 + 2 = 6 | 25 + 6 + 20 - 1 = 50
54 | 19 + 2 = 21 | 4 + 4 = 8 | 29 + 8 + 18 - 1 = 54
72 | 27 + 4 = 31 | 4 + 2 = 6 | 37 + 12 + 24 - 1 = 72
75 | 25 + 2 = 27 | 2 + 1 = 3 | 30 + 6 + 40 - 1 = 75
80 | 32 + 3 = 35 | 3 + 1 = 4 | 39 + 10 + 32 - 1 = 80
96 | 38 + 7 = 45 | 4 + 4 = 8 | 53 + 12 + 32 - 1 = 96
98 | 41 + 3 = 44 | 5 + 2 = 7 | 51 + 6 + 42 - 1 = 98
100 | 42 + 4 = 46 | 4 + 2 = 6 | 52 + 9 + 40 - 1 = 100
108 | 44 + 8 = 52 | 5 + 4 = 9 | 61 + 12 + 36 - 1 = 108
112 | 48 + 3 = 51 | 3 + 1 = 4 | 55 + 10 + 48 - 1 = 112
In this way A361098 presents a sort of capstone to the concept of constitutive state counting functions.
A359929: V5105: Irregular table of symmetric semidivisors k ¦¦ n for strong tantus n.
29 March 2023 {12, 18, 24, 18, 36, 20, 40, 12, 24, 36, 48, 48, 54, 45, 50, 60, 18, 36, 54, 72, 28, 56, 40, 80, 24, 48, 72, 96, 98, 90, 84, 75, 54, 96, 108, 63, 60, 90, 120, 50, 100, …}
Irregular triangle read by rows, where row n lists k < t such that rad(k) = rad(t) but k does not divide t, where t = A360768(n) and rad(k) = A7947(k).
Table of some of the first rows of the sequence, showing both even and odd b(n) = A360768(n) with both a single and multiple terms in the row:
n b(n) row n of this sequence
---------------------------------
1 18 12;
2 24 18;
3 36 24;
4 48 18, 36;
5 50 20, 40;
6 54 12, 24, 36, 48;
...
8 75 45;
...
18 135 75;
...
23 162 12, 24, 36, 48, 72, 96, 108, 144;
...
56 375 45, 135, 225;
57 378 84, 168, 252, 294, 336;
58 384 18, 36, 54, 72, 108, 144, 162, 216, 288, 324, etc.
A359382: V5106: Number of symmetric semidivisors k < n for strong tantus n,: i.e., ξ₉(A360768(n)).
29 March 2023 {1, 1, 1, 2, 2, 4, 2, 1, 1, 1, 4, 2, 2, 4, 1, 1, 1, 1, 3, 1, 3, 2, 8, 1, 2, 1, 7, 2, 1, 2, 5, 2, 1, 1, 3, 3, 1, 6, 1, 1, 5, 5, 4, 5, 1, 1, 4, 8, 3, 3, 1, 2, 1, 4, 2, 3, 5, 10, …}
This sequence contains nonzero values from A355432.
Chart below shows k < b(n) such that rad(k) = rad(b(n)), yet k does not divide b(n) (i.e., k ⑨b(n)). Sequence b(n) = A360768(n):
n b(n) List of k a(n)
------------------------------
1 18 12 1
2 24 18 1
3 36 24 1
4 48 18, 36 2
5 50 20, 40 2
6 54 12, 24, 36, 48 4
...
A361487: V7007: Odd strong tantus numbers: odd numbers k that are neither prime powers nor squarefree, such that k/rad(k) ≥ q, where prime q = A119288(k).
29 March 2023 {75, 135, 147, 189, 225, 245, 363, 375, 405, 441, 507, 525, 567, 605, 675, 735, 825, 845, 847, 867, 875, 891, 945, 975, 1029, 1053, 1083, 1089, 1125, …}
This sequence is { A5408 ∩ A360768 } ⊂ A126706.
Chart below shows k < a(n) such that rad(k) = rad(n), yet k does not divide n (i.e., k ⑨ n):
75 | 45 .
135 | . . 75 . .
147 | . 63 . . . .
189 | . . . . . . 147 . . .
a(n) 225 | . . . . . 135 . . . . . .
245 | . . . . . . . . . 175 . . .
363 | . . . 99 . . . . . . . . . . . . . 297
375 | 45 . . . . 135 . . . . . . 225 . . . . .
----------------------------------------------------------------------------
| 45 63 75 99 117 135 147 153 171 175 189 207 225 245 261 275 279 297
k in A360769
A362010: V6601: S₄₂: Numbers m such that 1 < (m, 42) < m and rad(m) ∤ 42.
4 April 2023 {10, 15, 20, 22, 26, 30, 33, 34, 35, 38, 39, 40, 44, 45, 46, 50, 51, 52, 57, 58, 60, 62, 66, 68, 69, 70, 74, 75, 76, 77, 78, 80, 82, 86, 87, 88, 90, …}.
This sequence is { ℕ \ { A108319 ∪ A206547 } }.
Image below shows k mod 42 : k ∈ A362010 in black, k ∈ T₄₂ = A206547 in white, and k ∈ R₄₂ = A108319 in color. k = 1 is shown in purple, k | 42 in red, otherwise, k ¦ 42 in gold.
A362011: V6602: S₇₀: Numbers m such that 1 < (m, 70) < m and rad(m) ∤ 70.
4 April 2023 {6, 12, 15, 18, 21, 22, 24, 26, 30, 34, 36, 38, 42, 44, 45, 46, 48, 52, 54, 55, 58, 60, 62, 63, 65, 66, 68, 72, 74, 75, 76, 77, 78, 82, 84, 85, 86, 88, 90, …}.
This sequence is { ℕ \ { A108513 ∪ A235583 } }.
Image below shows k mod 70 : k ∈ A362011 in black, k ∈ T₇₀ = A235583 in white, and k ∈ R₇₀ = A108513 in color. k = 1 is shown in purple, k | 70 in red, otherwise, k ¦ 70 in gold.
A362012: V6603: S₁₀₅: Numbers m such that 1 < (m, 105) < m and rad(m) ∤105.
4 April 2023 {6, 10, 12, 14, 18, 20, 24, 28, 30, 33, 36, 39, 40, 42, 48, 50, 51, 54, 55, 56, 57, 60, 65, 66, 69, 70, 72, 77, 78, 80, 84, 85, 87, 90, 91, 93, 95, 96, 98, 99, 100, …}.
This sequence is { ℕ \ { A108347 ∪ A236206 } }.
Image below shows k mod 105 : k ∈ A362012 in black, k ∈ T₁₀₅ = A236206 in white, and k ∈ R₁₀₅ = A108347 in color. k = 1 is shown in purple, k | 105 in red, otherwise, k ¦ 105 in gold.
A362003: V5413: Varius numbers m such that k − m² < m, where k is the smallest number exceeding m² such that rad(k) | m.
5 April 2023 {42, 66, 78, 362, 1086, 1122, 1254, 1794, 1810, 1846, 1974, 2534, 2730, 3318, 3982, 4890, 5538, 5590, 6006, 6214, 9230, 12922, …}.
This sequence is { m : A362045(n) − m² < m and m ∈ A120944 }.
Most small squarefree m have k − m² > m. For prime m = p, k = p³, hence (p³ − p²) > p.
A362044: V5411: Largest k such that k < m² and k ¦ m, where varius m = A120944(n) = V6(n).
5 April 2023 {32, 80, 128, 135, 343, 352, 512, 864, 891, 1088, 875, 1216, 1053, 1728, 2048, 2187, 1375, 2187, 2048, 2048, 3125, 4224, 2187, …}.
The largest k such that k < p² such that p is prime and rad(k) | p is p itself.
A362045: V5412: Smallest k such that k > m² and k ¦ m, where varius m = A120944(n) = V6(n).
5 April 2023 {48, 125, 224, 243, 567, 512, 832, 960, 1331, 2048, 1715, 2048, 2187, 1792, 2944, 4131, 3125, 4617, 3712, 3968, 8125, 4374, 5589, 5000, 8192, …}.
The smallest k such that k > p² such that p is prime and rad(k) | p is p³.
(A362127-A362136, 10 April 2023, to be outlined soon).
A360529: V3103: Smallest k > A024619(n) such that rad(k) = rad(A024619(n)).
1 May 2023 {12, 20, 18, 28, 45, 24, 40, 63, 44, 36, 52, 56, 60, 99, 68, 175, 48, 76, 117, 50, 84, 88, 75, 92, 54, 80, 153, 104, 72, 275, 98, 171, …}.
This is to say, the smallest k > m that is strongly regular to m, where m is the n-th non-prime power.
We say that k is n-regular if and only if rad(k) | rad(n); k and n are strongly regular (or coregular) if and only if rad(k) = rad(n). We define the sequence R of n-regular numbers to be the tensor product of prime divisor power ranges of n. It is clear that R pertains to the squarefree kernel ϰ = rad(n), and so we may instead write Rϰ. Now we may produce the sequence of strongly n-regular numbers via ϰRϰ. This sequence begins with squarefree ϰ, followed by distinct mϰ,where m is ϰ-regular, in order of m. These mϰ, m > 1 are tantus (i.e., in A126706).
Define function f(x) to be that function that finds n in ϰRϰ and supplies its successor k as output.
This sequence is f(x) ↦ A024619.
The sequence is a permutation of tantus numbers.
We note that f(p) = p², and f(pε) = p(ε+1). Also, for m such that ω(m) = 1, m | k.
Squarefree m implies a(n) = lpf(m)×m.
A360719: V3102: Largest k < A126706(n) such that rad(k) = rad(A126706(n)).
1 May 2023 {6, 12, 10, 18, 14, 24, 20, 22, 15, 36, 40, 26, 48, 28, 30, 21, 34, 54, 45, 38, 50, 42, 44, 60, 46, 72, 56, 33, 80, 52, 96, …}.
This is to say, the largest k < m that is strongly regular to m, where m is the n-th tantus number.
We say that k is n-regular if and only if rad(k) | rad(n); k and n are strongly regular (or coregular) if and only if rad(k) = rad(n). We define the sequence R of n-regular numbers to be the tensor product of prime divisor power ranges of n. It is clear that R pertains to the squarefree kernel ϰ = rad(n), and so we may instead write Rϰ. Now we may produce the sequence of strongly n-regular numbers via ϰRϰ. This sequence begins with squarefree ϰ, followed by distinct mϰ,where m is ϰ-regular, in order of m. These mϰ, m > 1 are tantus (i.e., in A126706).
Define function f(x) to be that function that finds n in ϰRϰ and supplies its predecessor k as output.
This sequence is f(x) ↦ A126706.
The sequence is a permutation of non prime powers A024619.
We note that f(p²) = p, and f(pε) = p(ε−1). Also, for m such that ω(m) = 1, k | m.
A362041: V3100: a(0) = 1; for n > 0, a(n) is the largest k < A013929(n) such that rad(k) = rad(A013929(n)).
1 May 2023 {1, 2, 4, 3, 6, 8, 12, 10, 18, 5, 9, 14, 16, 24, 20, 22, 15, 36, 7, 40, 26, 48, 28, 30, 21, 32, 34, 54, 45, 38, 50, 27, 42, 44, 60, 46, 72, …}.
This is to say, the largest k < m that is strongly regular to m, where m is the n-th nonsquarefree number.
We say that k is n-regular if and only if rad(k) | rad(n); k and n are strongly regular (or coregular) if and only if rad(k) = rad(n). We define the sequence R of n-regular numbers to be the tensor product of prime divisor power ranges of n. It is clear that R pertains to the squarefree kernel ϰ = rad(n), and so we may instead write Rϰ. Now we may produce the sequence of strongly n-regular numbers via ϰRϰ. This sequence begins with squarefree ϰ, followed by distinct mϰ,where m is ϰ-regular, in order of m. These mϰ, m > 1 are tantus (i.e., in A126706) for varius ϰ in A120944, or multus (i.e., in A246547) if ϰ = p prime.
Define function f(x) to be that function that finds n in ϰRϰ and supplies its predecessor k as output.
This sequence is f(x) ↦ A013929.
The sequence is a permutation of natural numbers.
We note that f(p²) = p, and f(pε) = p(ε−1). Also, for m such that ω(m) = 1, k | m.
A363101: V7004: Even tantus numbers, i.e., A5483 ∩ A126706.
19 May 2023 {12, 18, 20, 24, 28, 36, 40, 44, 48, 50, 52, 54, 56, 60, 68, 72, 76, 80, 84, 88, 90, 92, 96, 98, 100, 104, 108, 112, 116, 120, 124, 126, 132, 136, 140, 144, …}.
A362844: V5108: a(n) is the largest k < A360768(n) such that rad(k) = rad(A360768(n)) and n mod k ≠ 0, where rad(n) = A7947(n).
20 May 2023 {12, 18, 24, 36, 40, 48, 54, 45, 50, 60, 72, 56, 80, 96, 98, 90, 84, 75, 108, 63, 120, 100, 144, 126, 150, 147, 162, 112, 132, 160, 192, 196, 135, 156, 180, 176, 175, 200, …}.
Largest nondivisor less than m = A360768(n) that shares the same squarefree kernel as m.
Largest symmetric m-semidivisor k, i.e., nondivisor k coregular to m.
a(n) ∈ A126706, not a permutation of tantus numbers A126706.
A362432: V5109: a(n) is the smallest k > A126706(n) such that rad(k) = rad(A126706(n)) and k mod n ≠ 0, where rad(n) = A7947(n).
20 May 2023 {18, 24, 50, 36, 98, 48, 50, 242, 75, 54, 80, 338, 72, 98, 90, 147, 578, 96, 135, 722, 100, 126, 242, 120, 1058, 108, 112, 363, 160, 338, 144, 196, 1682, 507, 150, 1922, …}.
Let m = A126706(n) and r = rad(m).
Smallest number k greater than m that shares the same squarefree kernel as m, yet does not divide m.
a(n) ∈ A126706, not a permutation of tantus numbers A126706.
k/r and m/r are coprime.
a(n) < m², since k/m < r.
A363216: V8004: Even plenus numbers, i.e., even powerful numbers that are not powers of primes.
21 May 2023 {36, 72, 100, 108, 144, 196, 200, 216, 288, 324, 392, 400, 432, 484, 500, 576, 648, 676, 784, 800, 864, 900, 968, 972, 1000, 1152, 1156, 1296, …}.
This sequence is { A286708 ∩ A005843 } = { A1694 ∩ A363101 }.
Subset of A1694, A126706, and A363101.
A363217: V8005: Odd plenus numbers, i.e., odd powerful numbers that are not powers of primes.
21 May 2023 {225, 441, 675, 1089, 1125, 1225, 1323, 1521, 2025, 2601, 3025, 3087, 3249, 3267, 3375, 3969, 4225, 4563, 4761, 5625, 5929, 6075, 6125, 7225, 7569, 7803, 8281, …}.
This sequence is { A286708 ∩ A005408 } = { A1694 ∩ A360769 }.
Subset of A001694, A062739, A126706, and A360769.
A363234: Least number divisible by the first n primes whose factorization into maximal prime powers, if ordered by increasing prime divisor, then has these prime power factors in decreasing order.
23 May 2023 {1, 2, 12, 720, 151200, 4191264000, 251727315840000, 1542111744113740800000, 10769764221549079560253440000000, …}.
a(n) is the least number in A347284 divisible by prime(n).
a(n) is the smallest positive integer divisible by prime(n) and p(j)ε(j) > p(j+1)ε(j+1) where ε(k) is the valuation of p(k) in a(n) and 1 ≤ j < n (D. Corneth).
A363250: Let ℓ = ω(a(n)), p(k) be the k-th prime, and ε(k) be the maximal multiplicity such that p(k)ε(k) | a(n); a(0) = 1; For n > 0, a(n) = p(k) × ∏_{j=1..k} p(j)ε(j) where k is the index of the greatest power factor p(k)ε(k) such that p(k−1)ε(k−1) > p(k)(ε(k)+1).
21 May 2023 {1, 2, 4, 12, 8, 24, 16, 48, 144, 720, 32, 96, 288, 1440, 864, 4320, 21600, 151200, 64, 192, 576, 2880, 1728, 8640, 43200, 302400, 128, 384, 1152, 5760, 3456, 17280, 86400, 604800, 10368, 51840, 259200, 1814400, 256, …}.
A361376: Rewrite A129912(n), a product of distinct primorials P(i) = A2110(i) instead as a sum of powers 2(i−1).
9 June 2023 {0, 1, 2, 3, 4, 5, 6, 8, 7, 9, 10, 16, 11, 17, 12, 13, 18, 19, 32, 14, 33, 20, 15, 21, 34, 35, 22, 24, 64, 23, 36, 25, 65, 37, 26, 66, 38, 27, 67, 40, 128, 39, 41, 28, 68, 129, …}.
Sort n expressed as a sum of distinct powers 2k according to interpretation as a product of distinct primorials P(k+1).
Permutation of nonnegative numbers.
Let S(n) be the set of indices of primorials P(i), reverse sorted, such that A129912(n) = Product_{k=1..m} S(n,k), where m = | S(n) |. Then a(n) = Sum_{k=1..m} 2(S(n,k)−1).
n A129912(n) S(n) a(n) A272011(a(n))
-----------------------------------------
1 1 0
2 2 1 1 0
3 6 2 2 1
4 12 2,1 3 1,0
5 30 3 4 2
6 60 3,1 5 2,0
7 180 3,2 6 2,1
8 210 4 8 3
9 360 3,2,1 7 2,1,0
10 420 4,1 9 3,0
11 1260 4,2 10 3,1
12 2310 5 16 4
13 2520 4,2,1 11 3,1,0
14 4620 5,1 17 4,0
15 6300 4,3 12 3,2
16 12600 4,3,1 13 3,2,0
17 13860 5,2 18 4,1
18 27720 5,2,1 19 4,1,0
19 30030 6 32 5
...
Plot terms S(n) = A272011(a(n)) at (x,y) = (n, S(n,k)) for n = 1..960, 8× vertical magnification.
A363235: a(0) = 1; let ε be the largest multiplicity such that pε | a(n); for n>0, a(n) = ∑_{j=1..k} 2(ε(j)−1) where k is the index of the greatest power factor p(k)ε(k) such that p(k−1)ε(k−1) > p(k)ε(k+1).
9 June 2023 {0, 1, 2, 3, 4, 5, 8, 9, 10, 11, 16, 17, 18, 19, 20, 21, 22, 23, 32, 33, 34, 35, 36, 37, 38, 39, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 128, 129, 130, 131, 132, 133, 134, 135, …}.
A binary compactification of A363250, this sequence rewrites A363250(n) = Product_{i=1..ω(a(n))} p(i)ε(i) instead as Sum_{i=1..ω(a(n))} ε(i)−1.
Table relating this sequence to A363250.
b(n) = A363250(n), f(n) = A067255(n), g(n) = A272011(n).
n b(n) f(b(n)) a(n) g(a(n))
------------------------------------
1 1 0 0 -
2 2 1 1 0
3 4 2 2 1
4 12 2,1 3 1,0
5 8 3 4 2
6 24 3,1 5 2,0
7 16 4 8 3
8 48 4,1 9 3,0
9 144 4,2 10 3,1
10 720 4,2,1 11 3,1,0
11 32 5 16 4
12 96 5,1 17 4,0
13 288 5,2 18 4,1
14 1440 5,2,1 19 4,1,0
15 864 5,3 20 4,2
16 4320 5,3,1 21 4,2,0
17 21600 5,3,2 22 4,2,1
18 151200 5,3,2,1 23 4,2,1,0
19 64 6 32 5
...
Therefore, a(18) = 23 = 2⁴ + 2² + 2¹ + 2⁰ since b(18) = 151200 = 2⁵ × 3³ × 5² × 7¹.
The sequence is a series of intervals that begin as follows:
0
1
2..3
4..5
8..11
16..23
32..39
64..75
128..139 144..151
256..267 272..279
512..523 528..535 544..559
1024..1035 1040..1047 1056..1071
2048..2059 2064..2071 2080..2095 2112..2127
...
Plot terms S(n) = A272011(a(n)) at (x,y) = (n, S(n,k)) for n = 1..960, 8× vertical magnification.
A362227: a(n) = Product_{k=1..A120(n)} p(k)(S(n,k)−1), where set S(n,k) = row n of A272011 and A120(n) is the binary weight of n.
9 June 2023 {1, 2, 4, 12, 8, 24, 72, 360, 16, 48, 144, 720, 432, 2160, 10800, 75600, 32, 96, 288, 1440, 864, 4320, 21600, 151200, 2592, 12960, 64800, 453600, 324000, 2268000, 15876000, 174636000, …}.
In other words, let S(n) contain place values of 1's in the binary expansion of n, ordered greatest to least, where S(n,1) = ⌊log₂(n+1)⌋ = A523(n+1) and the remaining terms in S strictly decrease. This sequence reads S(n,k)+1 instead as a multiplicity of prime(k) so as to produce a number with strictly decreasing prime exponents.
This sequence, sorted, is A087980.
a(2k) = 2(k+1).
a(2k−1) = A6939(k−1).
This sequence seen as an irregular triangle T(m,j) delimited by 2m where j = 1..2(m−1) for m > 0.
1;
2;
4, 12;
8, 24, 72, 360;
16, 48, 144, 720, 432, 2160, 10800, 75600;
...
T(m,1) = 2m.
T(m, 2(m−1)) = A6939(m) for m > 0.
n S(n) a(n) A067255(a(n))
------------------------------------
1 0 2 1
2 1 4 2
3 1,0 12 2,1
4 2 8 3
5 2,0 24 3,1
6 2,1 72 3,2
7 2,1,0 360 3,2,1
8 3 16 4
9 3,0 48 4,1
10 3,1 144 4,2
11 3,1,0 720 4,2,1
12 3,2 432 4,3
13 3,2,0 2160 4,3,1
14 3,2,1 10800 4,3,2
15 3,2,1,0 75600 4,3,2,1
16 4 32 5
...
A363537: Rewrite A087980(n) = Product_{i=1..m} p(i)ε(i) instead as Sum_{i=1..m} 2(i−1), where m = ω(A087980(n)) = A1221(A087980(n)).
9 June 2023 {0, 1, 2, 4, 3, 8, 5, 16, 9, 32, 6, 17, 64, 10, 33, 128, 18, 7, 65, 12, 256, 34, 11, 129, 20, 512, 66, 19, 257, 36, 1024, 13, 130, 24, 35, 513, 68, 2048, 21, 258, 40, 67, 1025, 132, 4096, …}.
Permutation of nonnegative numbers.
Rewriting nonnegative numbers n = Sum_{i=1..A120(n)} 2i instead as Product_{i=1..A120(n)} p(i)(ε(i)+1) gives A362227.
a(2k) = 2(k−1) for k > 0.
a(A6939(k)) = 2(k−1) for k > 0.
Table relating this sequence to A087980, where b(n) = A087980(n), f(n) = A067255(n), g(n) = A272011(n), and a(n)_2 the binary expansion of a(n):
n b(n) f(b(n)) a(n) g(a(n)) a(n)_2
------------------------------------------
1 1 0 0
2 2 1 1 0 1
3 4 2 2 1 1.
4 8 3 4 2 1..
5 12 2,1 3 1,0 11
6 16 4 8 3 1...
7 24 3,1 5 2,0 1.1
8 32 5 16 4 1....
9 48 4,1 9 3,0 1..1
10 64 6 32 5 1.....
11 72 3,2 6 2,1 11.
12 96 5,1 17 4,0 1...1
13 128 7 64 6 1......
14 144 4,2 10 3,1 1.1.
15 192 6,1 33 5,0 1....1
16 256 8 128 7 1.......
17 288 5,2 18 4,1 1..1.
18 360 3,2,1 7 2,1,0 111
...
A363794: V3202: Smallest P(n)-regular k such that θ(k) ≥ θ(P(n+1)).
22 Jun 2023 {16, 72, 540, 6300, 92400, 1681680, 36756720, 921470550, 27886608750, 970453984500, 37905932634570 …}
Let R = θ(P(n)) = A010846(A2110(n)) = A363061(n).
Let S(n) be the sorted tensor product of prime power ranges {p(i)ε : i ≤ n, ε ≥ 0}, e.g., S(1) = A79, S(2) = A3586, S(3) = A051037, etc.
Let T(n) = A2110(n) × S(n). Note that S(1) = T(1) since ω(A2110(1)) = 1.
Let S(n,i) be the i-th term in S(n).
Then a(n) is the smallest S(n,i), i ≥ R, such that S(n,i) is also in T. Equivalently, a(n) is the smallest S(n,i), i ≥ R, such that rad(S(n,i)) = A2110(n), where rad(n) = A7947(n).
Table showing the relationship of a(n) to r(P(n)) = A0363061(n), with p(n) = prime(n), P(n+1) = A2110(n+1), r(a(n)) = A010846(a(n)), and j the index such that S(r(a(n))) = T(j) = a(n). a(n) = m*P(n)
n p(n) P(n+1) a(n) r(P(n)) r(a(n)) j m
--------------------------------------------------------------
1 2 6 16 5 5 4 8
2 3 30 72 18 18 8 12
3 5 210 540 68 69 13 18
4 7 2310 6300 283 290 22 30
5 11 30030 92400 1161 1165 29 40
6 13 510510 1681680 4843 4848 42 56
7 17 9699690 36756720 19985 19994 53 72
8 19 223092870 921470550 83074 83435 68 95
9 23 6469693230 27886608750 349670 351047 89 125
10 29 200560490130 970453984500 1456458 1457926 107 150
A363844: V6111: ξt(P(n)) = V61 ↦ V0111 = V243823 ↦ A2110. Number of semitotatives of primorials.
23 Jun 2023 {0, 0, 0, 5, 95, 1548, 23110, 413508, 8020826, 186514437, 5447473481, 169902931273, 6317112341154, 260105450523376, 11228680152402376, 529602052783103298, …}
A363595: Recursive product of aliquot divisors of n.
10 Jul 2023 {1, 1, 1, 2, 1, 6, 1, 16, 3, 10, 1, 1728, 1, 14, 15, 2048, 1, 5832, 1, 8000, 21, 22, 1, 4586471424, 5, 26, 81, 21952, 1, 24300000, 1, 67108864, 33, 34, 35, 101559956668416, …}.
a(n) >= A7956(n).
a(p) = 1 for prime p.
a(p²) = p.
a(pε) = A295(ε).
a(p×q) = p×q for primes p, q, p < q.
A7947(n) | a(n) for n with ω(n) > 2.
A362266: Triangle read by rows: T(n,k) = LCM({pj−1 : j = 1…n})/(pk−1) for prime p.
10 Jul 2023 {1, 2, 1, 4, 2, 1, 12, 6, 3, 2, 60, 30, 15, 10, 6, 60, 30, 15, 10, 6, 5, 240, 120, 60, 40, 24, 20, 15, 720, 360, 180, 120, 72, 60, 45, 40, 7920, 3960, …}.
T(n, 1) = A058254(n), T(n,k) = A058254(n)/A006093(k).
A363082: V0703: “Thin” tantus numbers k: k ∈ A126706 such that q×r > k, where q = A053669(k) is the smallest prime that does not divide k and r = A7947(k) is the squarefree kernel.
10 Jul 2023 {12, 18, 20, 24, 28, 44, 52, 60, 68, 76, 84, 90, 92, 116, 120, 124, 126, 132, 140, 148, 150, 156, 164, 168, 172, 180, 188, 198, 204, 212, 220, 228, 234, 236, 244, 260, …}.
This sequence is A126706 \ A360765.
A363596: a(n) = (∏_{k=1…π(n+1)} prime(k)⌊n/(prime(k)−1)⌋)/(n+1)!.
10 Jul 2023 {1, 1, 2, 1, 6, 2, 12, 3, 10, 2, 12, 2, 420, 60, 24, 3, 90, 10, 420, 42, 660, 60, 360, 30, 3276, 252, 56, 4, 120, 8, 3696, 231, 3570, 210, 36, 2, 103740, …}.
The table below relates b(n) = A091137(n) to a(n), with (n+1)! × a(n) = k! × m = b(n), where k! is the largest factorial that divides b(n).
n A067255(b(n)) (n+1)!*a(n) k! * m
---------------------------------------
0 0 1! * 1 1! * 1
1 1 2! * 1 2! * 1
2 2.1 3! * 2 3! * 2
3 3.1 4! * 1 4! * 1
4 4.2.1 5! * 6 6! * 1
5 5.2.1 6! * 2 6! * 2
6 6.3.1.1 7! * 12 7! * 12
7 7.3.1.1 8! * 3 8! * 3
8 8.4.2.1 9! * 10 10! * 1
9 9.4.2.1 10! * 2 10! * 2
10 10.5.2.1.1 11! * 12 12! * 1
11 11.5.2.1.1 12! * 2 12! * 2
12 12.6.3.2.1.1 13! * 420 15! * 2
13 13.6.3.2.1.1 14! * 60 15! * 4
14 14.7.3.2.1.1 15! * 24 15! * 24
15 15.7.3.2.1.1 16! * 3 16! * 3
16 16.8.4.2.1.1.1 17! * 90 18! * 5
...
A364702: Numbers k in A361098 that are not divisible by A7947(k)².
V73: Nonplenus panstitutive numbers, V74 \ V8.
3 August 2023: {48, 50, 54, 75, 80, 96, 98, 112, 135, 147, 160, 162, 189, 192, 224, 240, 242, 245, 250, 252, 270, 294, 300, 320, 336, 338, 350, 352, 360, …}.
A361098 \ A286708.
In classmaker and classlabel level 3, numbers of this subclass are colored blue, but appear purple (RGB 0x800080) in level 4.
A363597: Union of prime powers and nonsquarefree numbers.
V0106: Nonvarius numbers.
15 August 2023: {1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 31, 32, 36, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 60, …}.
ℕ \ A120944.
Numbers that are prime powers pm, m ≥ 0, or products of multiple powers of distinct primes pm where at least 1 prime power pm is such that m > 1.
A364997: Numbers k neither squarefree nor prime power such that rad(k) × A119288(k) > k and rad(k) × A053669(k) < k. (Intersection of A360765 and A360767.)
V77: Thick-weak tantus numbers = V0701 ∩ V0702.
16 Aug 2023: {40, 45, 56, 63, 88, 99, 104, 117, 136, 152, 153, 171, 175, 176, 184, 207, 208, 232, 248, 261, 272, 275, 279, 280, 296, 297, 304, 315, 325, 328, 333, …}.
A126706 ⊃ A364997.
For a(n) = k, A355432(k) = 0 but A360543(k) > 0.
Terms in this sequence do not have nondivisor m < k such that rad(m) = rad(k), but do have m < k, gcd(m,k) > 1 such that both ω(k) > ω(m) and rad(m) | k. Simply, terms in this sequence do not have symmetric semidivisors but do have asymmetric semitotatives.
In classmaker and classlabel level 4, numbers of this subclass are colored hue(16/27, 1, 2/3), i.e., approximately RGB 0048ab.
A364998: Numbers k neither squarefree nor prime power such that rad(k) × A119288(k) ≤ k and rad(k) × A053669(k) > k. (Intersection of A360768 and A363082.)
V76: Thin-strong tantus numbers = V0700 ∩ V0703.
16 Aug 2023: {18, 24, 90, 120, 126, 150, 168, 180, 198, 234, 264, 306, 312, 342, 408, 414, 456, 522, 552, 558, 630, 666, 696, 738, 744, 774, 840, 846, 888, 954, 984, 990, 1032, …}.
A126706 ⊃ A364998.
For a(n) = k, A355432(k) > 0 but A360543(k) = 0.
Terms in this sequence have nondivisor m < k such that rad(m) = rad(k), but not m < k, gcd(m,k) > 1 such that both ω(k) > ω(m) and rad(m) | k. Simply, terms in this sequence have symmetric semidivisors but not asymmetric semitotatives.
In classmaker and classlabel level 4, numbers of this subclass are colored blue.
A364999: Numbers k neither squarefree nor prime power such that both rad(k) × A119288(k) > k and rad(k) × A053669(k) > k. (Intersection of A360767 and A363082.)
V75: Thin-weak tantus numbers = even minimally tantus numbers = V0701 ∩ V0703 = V70 \ V71.
16 Aug 2023: {12, 20, 28, 44, 52, 60, 68, 76, 84, 92, 116, 124, 132, 140, 148, 156, 164, 172, 188, 204, 212, 220, 228, 236, 244, 260, 268, 276, 284, 292, 308, 316, …}.
A126706 ⊃ A364999.
A355432(k) = A360543(k) = 0 for k = a(n).
For terms in this sequence, there exist neither nondivisor m < k such that rad(m) = rad(k), nor m < k, gcd(m,k) > 1 such that both ω(k) > ω(m) and rad(m) | k. This is to say, there are no m ≤ k : ∄ m ¦¦ k ∧ (m ◊¦ k ∨ m ◊| k). Simply, terms in this sequence have neither symmetric semidivisors nor asymmetric semitotatives.
In classmaker and classlabel level 4, numbers of this subclass are colored hue(16/27), i.e., approximately RGB 0070ff.
A364996: Union of A360767 and A363082.
V78: Nonpanstitutive tantus numbers = V7 \ V74 = V0701 ∪ V0703.
26 Aug 2023: {12, 18, 20, 24, 28, 40, 44, 45, 52, 56, 60, 63, 68, 76, 84, 88, 90, 92, 99, 104, 116, 117, 120, 124, 126, 132, 136, 140, 148, 150, …}.
A360767 ∪ A363082 = A126706 \ A351098 = ∪(A364997, A364998, A364999).
In classmaker and classlabel level 3, numbers of this subclass are colored hue(16/27), i.e., approximately RGB 0070ff.
A364919: a(0) = 1; a(n) is the least novel m such that rad(m) | A019565(n).
30 Aug 2023: {1, 2, 3, 4, 5, 8, 9, 6, 7, 14, 21, 12, 25, 10, 15, 16, 11, 22, 27, 18, 55, 20, 33, 24, 49, 28, 63, 32, 35, 40, 45, 30, 13, 26, 39, 36, 65, 50, 75, 48, 91, 52, 81, 42, 125, …}.
Let ϰ be a squarefree number and define Rϰ to be the set of numbers m such that rad(m) | ϰ.
For n > 0, a(n) is the smallest m in Rϰ such that a(j) ≠ m, j < n.
Conjecture: permutation of natural numbers.
A363084: Numbers k such that sqrt(A007947(k) − A007913(k)) is an integer m > 0.
5 Sep 2023: {4, 16, 18, 25, 64, 72, 100, 162, 180, 256, 288, 289, 294, 400, 507, 625, 648, 676, 720, 722, 1024, 1152, 1176, 1210, 1369, 1458, 1600, 1620, 2178, 2205, 2500, …}.
Let core(k) = A7913(k) and rad(k) = A7947(k).
Squarefree numbers k imply rad(k) − core(k) = k − k = 0.
Perfect squares k² such that rad(k) = m²+1 and k > 1 imply rad(k²) − core(k²) = (m²+1) − 1 = m², with integers k, m.
Generally, if there exists a minimal d such that d | k, k/d = m², and rad(k) − d = m², then k is in the sequence.
Subsets of this sequence include the sets of squares k² such that k is in A2496, A3592, and A089653, since A089653 contains both A2496 and A3592.
A365786 20230919 V8000
A365787 20230919 V8001
A365788 20230920 V3107
A365789 20230919 V8006
A364902 20230921 DS
A365899 20230928
A366103 20230929 DS
A365901 20231011 (recursively highly composite)
A365902 20231012
A366601 20231014 DS
A363924 20231024 V3150
A364225 20231024 V3111
A366926 20231028 PM
A367082 20231106 DS
A365324 20231115
A365413 20231115
A365642 20231115
A365656 20231117
A365435 20231119
A365745 20231210 V0087
A363280 20231212 V0212
A366786 20231216 V6001
A363814 20231218 V0704
A366854 20240101 V0082
A368749 20240104 DS
A366460 20230105 V0071
A365710 20240105 V1207
A366413 20240107 V0602
A368107 20240115
A367268 20240120 V0704
A368089 20240120 V0709
A369150 20240120 V0770
Drafts:
Over the next few months I will be entering “constitutive” sequences that result from the partitioning of tantus numbers (A126706) into blocks, that is, pairwise disjoint subsets. Since August 2023 I have fallen out of practice of writing mini-abstracts for such sequences, but will catch up early in 2024.
Sequences in the making:
The sequences CS, CV, DV, LV in this paper should be added in the near future.