V-Sequences

by Michael Thomas De Vlieger, updated 15 December 2023, St. Louis, Missouri. First published 18 May 2018.

Originally titled, “Mike’s Favorite Sequences”, this page intends to simplify and render in a concise manner those sequences that I have found most interesting or find frequent reference in my work. Some of these sequences have a “V-number” assigned to them that is more aligned to personal usage. The V-numbers are not intended to replace Sloane’s A-numbers (which are preferred over V-numbers), but better connect sequences I find myself using most often, as well as designate sequences I write before acceptance at OEIS. In the case where a sequence is not of general interest, the V-number serves as a ready handle.

Page to the end for more about S- and T-numbered sequences.

Over the past decade I have contributed over 300 original sequences to the OEIS. Many of the sequences I've contributed concern a couple elementary number-theoretical subjects I would consider secondary or tertiary to the well-known concepts of the divisor and the “totative”. The latter term refers to a number m < n such that gcd(m,n) = 1, i.e., m is coprime to n. Some sequences are tools by which we can “paraphrase” or abbreviate the notation of the prime decomposition of n. My interests have evolved from examination of arithmetic in various integer number bases. These interests have grown to focus on regular numbers 1 ≤ m ≤ n such that m | n^e with e ≥ 0, of which the divisor is a special case. Another focus is upon “neutral” numbers, i.e., nondivisors m in the cototient of n, of which there are precisely two species, the “semidivisor”, a nondivisor regular, and the “semitotative”, a nonregular composite m. Consequently, there are comparisons between semitotatives and semidivisors, and between divisors and semidivisors, etc. Many of the sequences I authored examine these topics.

Note, the Vinci Catalog (V-sequences) have extended beyond what is shown on this page. Click here for a textual list of V-sequences.

General

A001477: V0000: The nonnegative integers.
A000040: V0001: The primes.
A027746: V0002: The prime factors of n with multiplicity.
A002808: V0101: Composites.
A027748: V0200: The distinct prime factors of n.
A001221: V0201: (little) ω(n) = number of distinct prime factors of n
A001222: V0202: (big) Ω(n) = number of distinct prime factors of n counting multiplicity.
A020639: V1001: Least prime divisor of n.
A006530: V1002: Greatest prime divisor of n.
A002110: V0111: the primorials, products p_n# of the smallest n primes.
A060735: V0112: k × p_n# with 1 ≤ k < p_{(n + 1)}.
A007947: V0220: squarefree kernel of n: largest squarefree number k | n.
A055932: V0210: products of a contiguous set of the smallest primes, with multiplicity.
A025487: V0211: products of primorials (least integer of each prime signature).
—           : V0212: A025487 \ A002182.
A056808: V0213: A055932 \ A025487.
A124010: V0221: prime signature of n.
A280363: • V0230: floor(log_p(n)), with p the least prime that divides n.

Constitutive Classes of Number

A001694: V0003: Powerful numbers, i.e., products a² × b³ : a ≥ 1 ∧ b ≥ 1.
A246547: V0004: Multus numbers, i.e., composite prime powers.
A005117: V0005: Squarefree numbers.
A120944: V0006: Varius numbers, i.e., composite squarefree numbers.
A126706: V0007: Tantus numbers, i.e., { V3 \ V4 }.
A286708: V0008: Plenus numbers, i.e., products of multus numbers, V8 ⊂ V7.
A000961: V0009: The prime powers (including the empty product).
A360768: • V0700: Strong tantus numbers, i.e., tantus that foster k ⑨ n.
A360767: • V0701: Weak tantus numbers, V7 \ V70.
A360765: • V0702: Thick tantus numbers, i.e., tantus that foster k ③ n.
A363082: • V0703: Thin tantus numbers, V7 \ V72.
A366825: • V0070: Minimally tantus numbers: { k : k = p²m : Ω(m) = ω(m) > 1, m > 1 }.
A366460: • V0071: Odd minimally tantus numbers.
A303946: V0072: Non-valens A126706 \ A131605 = V7 \ V84.
A364702: • V0073: Nonplenus panstitutive A361098 \ A286708 = V74 \ V8.
A361098: • V0074: Panstitutive numbers, thick-strong tantus, V70 ∩ V72.
A364999: • V0075: Thin-weak tantus, V71 ∩ V73.
A364998: • V0076: Thin-strong tantus, V70 ∩ V73.
A364997: • V0077: Thick-weak tantus, V71 ∩ V72.
A364996: • V0078: Nonpanstitutive tantus, V7 \ V74.
A332785: • V0079: Nonplenus tantus numbers V7 \ V8.
A360769: • V0701: Odd tantus numbers.
A363101: • V0702: Even tantus numbers.
A361487: • V0703: Odd strong tantus numbers.
A303606: V0080: Planus numbers: Nonsquarefree powers of varius numbers.
A131605: • V0081: Perfect-power plenus (valens) numbers, A286708 ∩ A1597.
A366854: • V0082: Fortis numbers, perfect powers of tantus, A131605 ∩ A303606.
A365308: • V0083: Perfect powers of composite primorials, A100778 ∩ A303606.
A363216: • V8004: Even plenus numbers.
A363217: • V8005: Odd plenus numbers.
A365745: • V0087: A303606 \ A365308 = V80 \ V81.
A052486: V0088: Achilles numbers, A286708 \ A1597 = V8 \ V81 = V8 \ V2.
A359280: • V0089: Nonplanus numbers: V8 \ V80 = A286708 \ A303606.
A365745: • V0839: A303606 \ A365308 = V80 \ V81.
A006881: V0103 Squarefree Semiprimes.
A085971: * V0104 Nonmultus numbers. ℕ \ A246547. * (A085971 = ℕ \ {{1}∪A246547}.)
A013939: V0105 Nonsquarefree numbers. ℕ \ A5117.
A363597: V0106 Nonvarius numbers. ℕ \ A120944.
A303554: V0107 Nontantus numbers. ℕ \ A126706.
—           : V0108 Nonplenus numbers. ℕ \ A286708.
A024619: V0109 Non-prime powers. ℕ \ A961.
A013929: V0102: Nonsquarefree numbers.

Algorithms

A275055: • V1020 list of 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.
A275280: • V3020 list of n-regular 1 ≤ k ≤ n (such that that k | n^e with e ≥ 0), in order of appearance in a matrix of products that arranges the powers of prime divisors p of n along independent axes.

Exponentiation notations

A067255: V0320: “multiplicity notation”: exponents of primes p | n. Abbreviated MN(n). (See also A054841, a big-endian decimal concatenation of A067255.)
A287352: • V0321: “π-code” (pi-code), first differences of exponents of the prime divisors p arranged in order of magnitude of p from least to greatest. Abbreviated PC(n).
—            V0322: “zero-code”: ZC(A288813(n)), row n contains runs of zeros delimited by nonzero terms in A054841(A288813(n)), eliminating leading zeros. This assigns a “family code” to terms in A288813. Equivalent to subtracting 1 from every term in A067255(n) and eliminating leading zeros.
—            V0323: “primorial-code”: PR(A025487(n)), row n contains the indices of primorials p_n# that together produce A025487(n).

Divisors ■

A027750: V0010: list of divisors of n.
A000005: V0011: divisor counting function τ(n).
A051731: V0019: characteristic function of divisors of n.
A002182: V0012: indices of records in τ(n) (highly composite numbers).
A002183: V0013: records in τ(n).
A002201: V0014: superior highly composite numbers.
A004394: V0512: superabundant numbers.
A004490: V0514: colossally abundant numbers.
A183093: V1101: 1 less than Lean Divisor Counting Function τ₄(n).
A183094: V1102: 1 less than Plenary Divisor Counting Function τ₆(n).
A032742: V1400: A002201(n) is the product of first n terms of this sequence.
A073751: V0515: A004490(n) is the product of first n terms of this sequence.

Nondivisors

A173540: V0015: list of nondivisors of n.
A049820: V0016: nondivisor counting function.
A000040: V0017: indices of records in A049820 (the primes).
A040976: V0018: records in A049820 (prime p − 2).

Totatives ■

A038566: V0020: list of totatives of n (reduced residue system of n).
A000010: V0021: Euler totient function φ(n).
A054521: V0029: characteristic function of totatives of n.
A006005: V0022: indices of records in A000010 (the odd primes together with 1).
A006093: V0023: records in A000010 (prime p – 1).
A007310: V2500: T₆ = numbers m coprime to 6 = (2 × 3). 5-rough numbers.
A045572: V2501: T₁₀ = numbers m coprime to 10 = (2 × 5).
A162699: V2502: T₁₄ = numbers m coprime to 14 = (2 × 7).
A229829: V2503: T₁₅ = numbers m coprime to 15 = (3 × 5).
A160545: V2504: T₂₁ = numbers m coprime to 21 = (3 × 7).
A235933: V2505: T₃₅ = numbers m coprime to 35 = (5 × 7).
A007775: V2600: T₃₀ = numbers m coprime to 30 = (2 × 3 × 5). 7-rough numbers.
A206547: V2601: T₄₂ = numbers m coprime to 42 = (2 × 3 × 7).
A235583: V2602: T₇₀ = numbers m coprime to 70 = (2 × 5 × 7).
A236206: V2603: T₁₀₅ = numbers m coprime to 105 = (3 × 5 × 7).
A008364: V2650: T₂₁₀ = numbers m coprime to 210 = (2 × 3 × 5 × 7). 11-rough numbers.
A008365: V2651: T₂₃₁₀ = numbers m coprime to 2310 = (2 × 3 × 5 × 7 × 11). 13-rough numbers.
A008366: V2652: T₃₀₀₃₀ = numbers m coprime to 30030 = (2 × 3 × 5 × 7 × 11 × 13). 17-rough numbers.
A166061: V2653: T₅₁₀₅₁₀ = numbers m coprime to 510510 = (2 × 3 × 5 × 7 × 11 × 13 × 17). 19-rough numbers.
A166063: V2654: T_9699690 = numbers m coprime to 9,699,690 = (2 × 3 × 5 × 7 × 11 × 13 × 17 × 19). 23-rough numbers.

Totient ratio graph; plot squarefree m at (x,y) = (π(gpf(m)), φ(m)/m):

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 = A076512(m)/A109395(m).
A000040: V0001: The primes. (maximum m in column x)
A006094: V2101: Products of 2 successive primes. (m with the second greatest φ(m)/m in column x)
A070826: V2102: p_n#/2. (odd m with least φ(m)/m in column x)
A325236: • V2103: Squarefree k such that φ(k)/k − ½ is positive and minimal for k with gpf(k) = prime(n).
A307544: • V2105: T(n,k) = A087207(A307540(n,k)). (Binary encoding of A307540)
A325237: • V2107: Squarefree k such that ½ − φ(k)/k is positive and minimal for k with gpf(k) = prime(n).
A077017: V2108: (even m with least φ(m)/m in column x)
A306237: • V2109: p_n#/prime(n − 1). (m with the second least φ(m)/m in column x)
A002110: V0111: The primorials. (minimum m in column x).

Cototatives

A121998: V0025: list of cototatives of n.
A051953: V0026: cototient function.
A063985: V0027: indices of records in A051953.
A063986: V0028: records in A051953.

Regular ■

Numbers 1 ≤ k ≤ n such that k | n^e with e ≥ 0.
Note the divisor d is a special case of regular k such that d | n^e with 0 ≤ e ≤ 1.
Nondivisor regulars are called “semidivisors” of n, as they divide n^e with e > 1.
Here we call the smallest exponent e of n^e such that k | n^e the “richness” of n-regular k.
For k = 1, e = 0. For divisors d > 1, e = 1. For semidivisors k, e > 2.

A162306: V0030: list of 1 ≤ k ≤ n such that k | n^e with e ≥ 0.
A010846: V0031: regular counting function θ(n).
A008479: V0034: coregular counting function, | { k : k ≤ n, rad(k) = rad(n) } |
A304569: • V0039: characteristic function of regulars of n.
A244052: • V0032: indices of records in A010846. “Highly regular numbers.”
A244053: • V0033: records in A010846.
A279907: • V0035: richness e of 1 ≤ k ≤ n.
A280269: • V0036: richness e of regulars 1 ≤ k ≤ n.
A280274: • V0037: maximum richness e of regulars 1 ≤ k ≤ n.
A294306: • V0038: population of richnesses e of regulars 1 ≤ k ≤ n.
A362041: • V3100: PrevStronglyRegular(n).
A065642: V3101: NextStronglyRegular(n).
A360719: • V3102: PrevStronglyRegular(n) → V7.
A360529: • V3103: NextStronglyRegular(n) → V0109.
—           : • V3105: NextStronglyRegular(n) → V5.
—           : • V3106: NextStronglyRegular(n) → V6.
A003586: V3500: R₆ = numbers m regular to 6 = (2 × 3). 3-smooth numbers.
A003592: V3501: R₁₀ = numbers m regular to 10 = (2 × 5).
A003591: V3502: R₁₄ = numbers m regular to 14 = (2 × 7).
A003593: V3503: R₁₅ = numbers m regular to 15 = (3 × 5).
A003594: V3504: R₂₁ = numbers m regular to 21 = (3 × 7).
A003595: V3505: R₃₅ = numbers m regular to 35 = (5 × 7).
A051037: V3600: R₃₀ = numbers m regular to 30 = (2 × 3 × 5). 5-smooth numbers.
A108319: V3601: R₄₂ = numbers m regular to 42 = (2 × 3 × 7).
A108513: V3602: R₇₀ = numbers m regular to 70 = (2 × 5 × 7).
A108347: V3603: R₁₀₅ = numbers m regular to 105 = (3 × 5 × 7).
A002473: V3650: R₂₁₀ = numbers m regular to 210 = (2 × 3 × 5 × 7). 7-smooth numbers.
A051038: V3651: R₂₃₁₀ = numbers m regular to 2310 = (2 × 3 × 5 × 7 × 11). 11-smooth numbers.
A080197: V3652: R₃₀₀₃₀ = numbers m regular to 30030 = (2 × 3 × 5 × 7 × 11 × 13). 13-smooth numbers.
A080681: V3653: R₅₁₀₅₁₀ = numbers m regular to 510,510 = (2 × 3 × 5 × 7 × 11 × 13 × 17). 17-smooth numbers.
A080682: V3654: R_9699690 = numbers m regular to 9,699,690 = (2 × 3 × 5 × 7 × 11 × 13 × 17 × 19). 19-smooth numbers.
A080683: V3655: R_223092870 = numbers m regular to 223,092,870 = (2 × 3 × 5 × 7 × 11 × 13 × 17 × 19 × 23). 23-smooth numbers.

Richnesses:

Let “richness” ρ_n(m) = ε : m | n^ε ∧ ε ≥ 0 ∧ (m | n^ε ∧ m | n^δ, 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 | K^ε iff m | n^ε. Then R_K is the infinite list of m | K^ε : ε ≥ 0, as well as all m | n^ε : ε ≥ 0. We note that A3586: V3500: R₆ = 6-regular m, i.e., 3-smooth numbers. This function ρ_n(x) is described by Hardy & Wright (4th-8th ed.) Theorem 136.
Therefore, let M be the mappings of ρ_n across R_K 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 M_p = ℕ₀ = A1477. For n = p^ε, we have M_n = { ⌊ℕ₀/ε⌋ }. Hence, for n = p², M_n = { ⌊ℕ₀/2⌋ } = A4526, for n = p³, M_n = { ⌊ℕ₀/3⌋ } = A2264, for n = p⁴, M_n = { ⌊ℕ₀/4⌋ } = A2265, etc. (These sequences do not merit notes in the respective OEIS entries.)

A086415: V3700: { ρ₆ ↦ R₆ } = richnesses of 6-regular m ∈ A3586.
A352072: • V3701: { ρ₁₂ ↦ R₆ } = richnesses of 12-regular m ∈ A3586.
—           V3702: { ρ₁₈ ↦ R₆ } = richnesses of 18-regular m ∈ A3586.
—           V3703: { ρ₂₄ ↦ R₆ } = richnesses of 24-regular m ∈ A3586.
—           V3704: { ρ₃₆ ↦ R₆ } = richnesses of 36-regular m ∈ A3586.
A117920: V3710: { ρ₁₀ ↦ R₁₀ } = richnesses of 10-regular m ∈ A3592.
A352218: • V3711: { ρ₂₀ ↦ R₁₀ } = richnesses of 20-regular m ∈ A3592.
—           V3720: { ρ₁₄ ↦ R₁₄ } = richnesses of 14-regular m ∈ A3591.
—           V3725: { ρ₁₅ ↦ R₁₅ } = richnesses of 15-regular m ∈ A3593.
—           V3730: { ρ₂₁ ↦ R₂₁ } = richnesses of 21-regular m ∈ A3594.
—           V3735: { ρ₃₅ ↦ R₃₅ } = richnesses of 35-regular m ∈ A3595.
—           V3740: { ρ₃₀ ↦ R₃₀ } = richnesses of 30-regular m ∈ A051037.
A352219: • V3741: { ρ₆₀ ↦ R₃₀ } = richnesses of 60-regular m ∈ A051037.
—           V3742: { ρ₉₀ ↦ R₃₀ } = richnesses of 90-regular m ∈ A051037.
—           V3743: { ρ₁₂₀ ↦ R₃₀ } = richnesses of 120-regular m ∈ A051037.
—           V3749: { ρ₃₆₀ ↦ R₃₀ } = richnesses of 360-regular m ∈ A051037.
—           V3770: { ρ₂₁₀ ↦ R₂₁₀ } = richnesses of 210-regular m ∈ A2473.
—           V3770: { ρ₂₁₀ ↦ R₂₁₀ } = richnesses of 210-regular m ∈ A2473.
—           V3773: { ρ₂₅₂₀ ↦ R₂₁₀ } = richnesses of 2520-regular m ∈ A2473.
—           V3774: { ρ₅₀₄₀ ↦ R₂₁₀ } = richnesses of 5040-regular m ∈ A2473.
—           V3775: { ρ₂₃₁₀ ↦ R₂₃₁₀ } = richnesses of 2310-regular m ∈ A051038.

Proper regulars:

An n-regular k is proper iff n ∤ k. The proper divisors d | n : d < n are also proper regular numbers with regard to n.

A353383: • V3800: P₁₂ = { k ∈ M₁₂ : 12 ∤ k } = irregular table, positional transform of proper 12-regular numbers.
A353384: • V3810: P₂₀ = { k ∈ M₂₀ : 20 ∤ k } = irregular table, positional transform of proper 20-regular numbers.
A353385: •V3841: P₆₀ = { k ∈ M₆₀ : 60 ∤ k } = irregular table, positional transform of proper 60-regular numbers.
—           V3843: P₁₂₀ = { k ∈ M₁₂₀ : 120 ∤ k } = irregular table, positional transform of proper 120-regular numbers.
—           V3849: P₃₆₀ = { k ∈ M₃₆₀ : 360 ∤ k } = irregular table, positional transform of proper 360-regular numbers.
—           V3873: P₂₅₂₀ = { k ∈ M₂₅₂₀ : 2520 ∤ k } = irregular table, positional transform of proper 2520-regular numbers.
—           V3874: P₅₀₄₀ = { k ∈ M₅₀₄₀ : 5040 ∤ k } = irregular table, positional transform of proper 5040-regular numbers.

Study of “highly regular numbers” A244052 (2016-7 “Turbulent Candidates” paper):

A288784: • V3200: Necessary but insufficient condition.
A288813: • V3201: Turbulent candidates in A288784.
A289171: • V3210: “Depth-Distension” correlation for primorial(n).
—            V3220: rcf(A002110(i), m) − A010846(m) for m in A288813 (Deficit of rcf(m) versus rcf(n, m)).

Neutral ■

Nondivisors k in the cototient of n, here called “neutral”.
Numbers k > n can also be neutral; they neither divide nor are coprime to n.
There are exactly two species of neutral k.
The “semidivisor” k | n^e with e > 1; k is the product of primes p that also divide n. The semidivisor is necessarily composite; semidivisors k < n pertain only to composite n with ω(n) > 1.
The “semitotative” k does not divide any power n^e, integers. It is the necessarily composite product of at least one prime p | n and one prime q coprime to n.

A133995: V0040: list of neutrals k < n.
A045763: V0041: neutral counting function (ncf(n), ξ(n)).
A304571: • V0049: characteristic function of neutrals of n.
A300859: • V0042: indices of records in A045763.
A300914: • V0043: records in A045763.
A294492: • V0044: indices of records for ratio A045763(n)/n.
A295523: • V0045: nonprime numbers with A243822(n) ≥ A243823(n). (finite, 7 terms).
A300858: • V0046: A243823(n) − A243822(n), i.e., ξ_t (n) − ξ_d (n).
A046022: V4600: zeros in A300858 (i.e., 1, 4, and the primes).
A300860: • V4601: indices of records in A300858.
A300861: • V4602: records in A300858.

Semidivisor ■

A272618: • V0050: list of 1 ≤ k ≤ n such that k | n^e with e > 1.
A243822: • V0051: semidivisor counting function ξ_d (n).
A304570: • V0059: characteristic function of semidivisors of n.
A293555: • V0052: indices of records in A243822. “Highly semidivisible numbers.”
A293556: • V0053: records in A243822.
A299990: • V0054: A243822(n) − A000005(n) = ξ_d (n) − τ(n).
A355432: • V5101: Symmetric Semidivisor Counting Function ξ₉(n).
A360589: • V5102: Indices of records in ξ₉(n) V5102 ⊂ V0210.
A360912: • V5103: Records in ξ₉(n).
A359929: • V5105: Symmetric semidivisors k ⑨ A360768(n).
A359382: • V5106: ξ₉(A360768(n)).
—            • V5110: Symmetric semidivisor kernels u for squarefree ϰ ∈ V5 (i.e., A5117).
—            • V5111: Symmetric semidivisor kernels u for squarefree ϰ ∈ V5 (i.e., A5117).
—            • V5112: Symmetric semidivisor differences δ = v − u, δ | ϰ for squarefree ϰ ∈ V5 (i.e., A5117).
—            • V5115: Symmetric semidivisor kernel counting function.
—            • V5116: Symmetric semidivisor kernel counting function (across V6).
—            • V5119: Symmetric semidivisor kernel counting function (across V0111).
A361235: • V5201: Asymmetric (Mixed) Semidivisor Counting Function ξ₇(n).
A299991: • V5401: numbers with A243822(n) > A000005(n), i.e., more semidivisors than divisors.
A299992: • V5402: composites n with ω(n) > 1 for which A243822(n) < A000005(n), i.e., fewer semidivisors than divisors. (numbers n with ω(n) = 1, including prime n, have no semidivisors less than n).
A300155: • V5400: numbers with A243822(n) = A000005(n), i.e., same number of divisors and semidivisors.
A300156: • V5403: indices of records in A299990.
A300157: • V5404: records in A299990.
A289280: V5410: Smallest k such that k > n and k ¦ n.
A362044: • V5411: largest k such that k < m² and k ¦ m, where m = A120944(n) = V6(n).
A362045: • V5412: smallest k such that k > m² and k ¦ m, where m = A120944(n) = V6(n).
A362003: • V5413: Varius m such that k − m² < m, where k > m² and k ¦ m.

Semitotative ■

A272619: • V0060: list of semitotatives.
A243823: • V0061: semitotative counting function ξ_t (n).
A304572: • V0069: characteristic function of semitotatives of n.
A096014: • V0068: smallest number m = pq semicoprime to n, i.e., prime p | n, and q the smallest prime nondivisor of n, with a(1) = 2.
A292867: • V0062: indices of records in A243823. “Highly semitotative numbers.”
A292868: • V0063: records in A243823.
A294575: • V0064: semitotative-dominant numbers (2 ×ξ_t (n) > n).
A360480: • V6101: Symmetric Semitotative Counting Function ξ₁(n).
A360543: • V6201: Asymmetric (Mixed) Semitotative Counting Function ξ₃(n).
—            • V6300: Opaque semitotatives of n. { k : k ◊ n ∧ k ≤ n ∧ k ∤ (n²−1) }.
A360224: • V6301: Opaque semitotative counting function ξ_h (n).
A361080: • V6302: Highly opaque numbers.
A361081: • V6303: Records in V6301.
—            • V6304: Opaque-semitotative dominant numbers.
—            • V6305: Numbers with at least as many divisors as opaque semitotatives.
—            • V6306: Numbers with at least as many regular numbers k ≤ n as opaque semitotatives.
A294576: • V6401: odd semitotative-dominant numbers.
A295221: • V6400: semitotative-parity numbers (2 × ξ_t (n) = n).
A291989: • V6410: smallest k such that k > n and k ◊ n.
A330136: • V6500: S₆ = numbers m semicoprime to 6 = (2 × 3).
A105115: V6501: S₁₀ = numbers m semicoprime to 10 = (2 × 5).
A316991: • V6502: S₁₄ = numbers m semicoprime to 14 = (2 × 7).
A316992: • V6503: S₁₅ = numbers m semicoprime to 15 = (3 × 5).
A306999: • V6504: S₂₁ = numbers m semicoprime to 21 = (3 × 7).
A307589: • V6505: S₃₅ = numbers m semicoprime to 35 = (5 × 7).
A330137: • V6600: S₃₀ = numbers m semicoprime to 30 = (2 × 3 × 5).
A362010: • V6601: S₄₂ = numbers m semicoprime to 42 = (2 × 3 × 7).
A362011: • V6602: S₇₀ = numbers m semicoprime to 70 = (2 × 5 × 7).
A362012: • V6603: S₁₀₅ = numbers m semicoprime to 105 = (3 × 5 × 7).
A084891: V6650: S₂₁₀ = numbers m semicoprime to 210 = (2 × 3 × 5 × 7).
—            • V6651: S₂₃₁₀ = numbers m semicoprime to 2310 = (2 × 3 × 5 × 7 × 11).
—            • V6652: S₃₀₀₃₀ = numbers m semicoprime to 30,030 = (2 × 3 × 5 × 7 × 11 × 13).
—            • V6653: S₅₁₀₅₁₀ = numbers m semicoprime to 510,510 = (2 × 3 × 5 × 7 × 11 × 13 × 17).
—            • V6654: S_9699690 = numbers m semicoprime to 9,699,690 = (2 × 3 × 5 × 7 × 11 × 13 × 17 × 19).

Aspects of Constitutive Classes and Constitutive Subclasses.
—            • V0400: {4} = V4 ∩ V14 = A246547 ∩ V2182.
—            • V0401: V4 \ {V120 ∪ V402} = A246547 \ {A79 ∩ A1248}
A000079: • V0402: Powers of 2.
A000244: • V0403: Powers of 3.

A010722 : V0600: {6} = V6 ∩ V14 = V0111 ∩ V14 = A120944 ∩ A2182 = A2110 ∩ A2182.
A350352: V0601: V6 \ V103 = A120944 \ A6881.
A366413: V0602 = V6 \ V111 = A120944 \ A2110
—            : V0603 = V601 \ V111 = V602 \ V103 = V6 \ {V103 ∪ V111}.
A046388: V0611: Odd squarefree semiprimes.
A100484: V0612: Even squarefree semiprimes.
A007304: V0613: Sphenic numbers.
A046386: V0614: Products of 4 distinct primes.
A046387: V0615: Products of 5 distinct primes.
—            : V0616: Products of 6 distinct primes.
—            : V0617: Products of 7 distinct primes.
—            : V0618: Products of 8 distinct primes.

A366825: • V0700: Minimally tantus numbers: { k : k = p²m : Ω(m) = ω(m) > 1, m > 1 }.
A360769: • V0701: Odd tantus numbers.
A363101: • V0702: Even tantus numbers.
A361487: • V0703: Odd strong tantus numbers.
—           • V0704: V78 \ V0700 = A364996 \ A366825.
A365783: • V0705: V0220 ↦ V7 = squarefree kernel of tantus numbers.
A365784: • V0706: V7(n)/V0750(n).
A365785: • V0707: V6(k) = V0750(n).
A365790: • V0708: V31 ↦ V7 = regular counting function of tantus numbers.
A365791: • V0709: V34 ↦ V7 = coregular counting function of tantus numbers.
—            • V0730: A126706 ∩ A055932 = V7 ∩ V0210.
—            • V0731: A126706 ∩ A056808 = A126706 ∩ {A055932 \ A025487}
A364710: V0732: A126706 ∩ A025487 = V7 ∩ V0211.
—            • V0733: A126706 ∩ A363280 = A126706 ∩ {A025487 \ A002182}
—            • V0735: A126706 \ A055932 = V7 \ V0210.
—            • V0736: A126706 \ A025487 = V7 \ V0211.
—            • V0737: A126706 \ A2182.
—            • V0740: A361098 ∩ A055932. (V0742).
—            • V0741: A361098 ∩ A056808.
—            • V0742: A361098 ∩ A025487. (V0743).
—            • V0744: A361098 \ A055932.
—            • V0745: A361098 \ A025487.
—            • V0749: A361098 \ A286708 = V74 \ V8. (V0740).
—            • V0750: A364999 ∩ A055932. (V0752).
—            • V0751: A364999 ∩ A056808.
—            • V0752: A364999 ∩ A025487. (V0753).
—            • V0753: V75 ∩ V14 = V75 ∩ V12 = {12, 60}.
—            • V0754: A364999 \ A055932.
—            • V0755: A364999 \ A025487.
—            • V0760: A364998 ∩ A055932. (V0762).
—            • V0761: A364998 ∩ A056808.
—            • V0762: A364998 ∩ A025487. (V0763).
—            • V0763: V76 ∩ V14 = {24, 120, 180, 840, 1260, 1680, 27720}.
—            • V0764: A364998 \ A055932.
—            • V0765: A364998 \ A025487.
—            • V0766: V76 ∩ V14 = {120}.
—            • V0770: A364997 ∩ A055932. (V0772).
—            • V0771: A364997 ∩ A056808.
—            • V0772: A364997 ∩ A025487. (V0773).
—            • V0774: A364997 \ A055932.
—            • V0775: A364997 \ A025487.
—            • V0780: A364996 ∩ A055932. (V0782).
—            • V0781: A364996 ∩ A056808.
—            • V0782: A364996 ∩ A025487. (V0783).
—            • V0783: V78 ∩ V14 = {12, 24, 60, 120, 180, 840, 1260, 1680, 27720}.
—            • V0784: A364996 \ A055932.
—            • V0785: A364996 \ A025487.
—            • V0790: A332785 ∩ A055932 = V79 ∩ V0210.
—            • V0791: A332785 ∩ A056808 = {A126706 \ A286708} ∩ {A055932 \ A025487}
—            • V0792: A332785 ∩ A025487 = V79 ∩ V0211.
—            • V0793: A332785 ∩ A363280 = {A126706 \ A286708} ∩ {A025487 \ A002182}
—            • V0794: A332785 ∩ A2182 = V79 ∩ V12.
—            • V0795: A332785 ∩ A333655 = V79 ∩ {V12 \ V14}.
—            • V0796: A332785 ∩ A2201 = V79 ∩ V14.
—            • V0797: A332785 \ A055932 = V79 \ V0210.
—            • V0798: A332785 \ A025487 = V79 \ V0211.
—            • V0799: A332785 \ A2182 = V79 \ V12.

—            • V0830: A286708 ∩ A055932.
—            • V0831: A286708 ∩ A056808 = A286708 ∩ {A055932 \ A025487}
A364930: • V0832: A286708 ∩ A025487.
—            • V0833: A286708 \ A055932.
—            • V0834: A286708 \ A025487.
—            • V0835: A303606 ∩ A025487.
—            •V0836: A303606 \ A025487.
A365745: • V0839: A303606 \ A365308 = V80 \ V81.
—            • V0890: A359280 ∩ A055932 = V89 ∩ V0210.
—            • V0891: A359280 ∩ A056808 = V89 ∩ {V0210 \ V0211}.
—            • V0892: A359280 ∩ A025487 = V89 ∩ V0211.
—            • V0893: A359280 \ A055932 = V89 \ V0210.
—            • V0894: A359280 \ A025487 = V89 \ V0211.
A365786: • V0850: V0220 ↦ V8 = squarefree kernel of plenus numbers.
A365787: • V0855: V8(n)/V0850(n).
A365788: • V0856: V6(k) = V0850(n).
A365792: • V0858: V31 ↦ V8 = regular counting function of plenus numbers.
A365793: • V0859: V34 ↦ V8 = coregular counting function of plenus numbers.
A365789: • V0857: V0109(k) = V0850(n).
—            • V0860: V0220 ↦ V80 = squarefree kernel of planus numbers.
—            • V0865: V7(n)/V0750(n).
—            • V0868: V31 ↦ V80 = regular counting function of planus numbers.
—            • V0869: V34 ↦ V80 = coregular counting function of planus numbers.

On the largest primorial divisor p_ω(m)# of highly composite and superabundant m:

A108602: V1230: ω(m) for highly composite m (in A2182).
A305025: V1231: ω(m) for superabundant m (in A4394).
A340840: • V1200: Highly composite or superabundant numbers: union of A2182 and A4394.
A166981: V1201: Superabundant numbers that are highly composite.
A224078: V1202: Superior highly composite numbers that are colossally abundant. (finite: 449 terms)
A304234: • V1203: Superior highly composite numbers that are superabundant but not colossally abundant. (39 terms)
A304235: • V1204: Colossally abundant numbers that are highly composite, but not superior highly composite. (34 terms)
A338786: • V1205: Numbers in A166981 that are neither superior highly composite nor colossally abundant.
A301413: • V1240: A002182(n)/A002110(A108602(n)), i.e., m/p_ω(m)# for highly composite m.
A301414: • V1241: primitive values in A301413.
A305056: • V1245: A004394(n)/A002110(A001221(A004394(n))), i.e., m/p_ω(m)# for superabundant m.
A340014: • V1242: primitive values in A305056.
—            • V1300: Union of A2201 and A4490 (data).
A301416: • V1341: Numbers in A301413 that produce superior highly composite numbers when multiplied by some primorial.
A340137: • V1342: Numbers in A305056 that produce colossally abundant numbers when multiplied by some primorial.
—            • V1250: Union of A1241 and A1242 (A301414 ∪ A340014) (data).
—            • V1350: Union of A1341 and A1342 (A301416 ∪ A340137) (data).

Indices of Certain Subset Functions in their Supersets

A331119: • V0250: Indices of A025487(n) in A055932 (i.e., V0211(n) in V0210).
A331938: • V0251: Indices of A002110(n) in A055932 (i.e., V0111(n) in V0210).
A332034: • V0252: Indices of A002182(n) in A055932 (i.e., V0012(n) in V0210).
A332035: • V0253: Indices of A004394(n) in A055932 (i.e., V0512(n) in V0210).

A098719: V0260: Indices of A002110(n) in A025487 (i.e., V0111(n) in V0211).
A306802: • V0261: Indices of A002182(n) in A025487 (i.e., V0012(n) in V0211).
A293635: V0262: Indices of A004394(n) in A025487 (i.e., V0512(n) in V0211).
—            V0263: Indices of A166981(n) in A025487 (i.e., V1200(n) in V0211).
A332241: • V0264: Indices of A224078(n) in A025487 (i.e., V1201(n) in V0211).
—            V0265: Indices of A304234(n) in A025487 (i.e., V1202(n) in V0211).
—            V0266: Indices of A304235(n) in A025487 (i.e., V1203(n) in V0211).

—            V1210: Indices of A004394(n) in A002182 (i.e., V0512(n) in V0012), 0 if no position.
—            V1211: Indices of A166981(n) in A002182 (i.e., V1200(n) in V0012).
—            V1212: Indices of A224078(n) in A002182 (i.e., V1201(n) in V0012).
—            V1213: Indices of A304234(n) in A002182 (i.e., V1202(n) in V0012).
—            V1214: Indices of A304235(n) in A002182 (i.e., V1203(n) in V0012).

—            V1215: Indices of A002182(n) in A004394 (i.e., V0012(n) in V0512), 0 if no position.
—            V1216: Indices of A166981(n) in A004394 (i.e., V1200(n) in V0512).
—            V1217: Indices of A224078(n) in A004394 (i.e., V1201(n) in V0512).
—            V1218: Indices of A304234(n) in A004394 (i.e., V1202(n) in V0512).
—            V1219: Indices of A304235(n) in A004394 (i.e., V1203(n) in V0512).

—            V1220: Indices of A002201(n) in A002182 (i.e., V0014(n) in V0012).
—            V1225: Indices of A004490(n) in A004394 (i.e., V0514(n) in V0512).

S- and T-Number Sequences.

Many with whom I have collaborated are aware of S-numbers that I designate that append the ISO-8601 date (8 digits). These are temporary reference tags that are supplanted by OEIS A-numbers upon acceptance. If they are connected to constitutive matters then they may also get a V-number. At times I also issue T-numbers, the reason being that I can assign the same date to a T number if it was already deployed to the S-number. I do not have a record of S- or T-numbers I have already designated. An example: S20230228 was designated V74 and then accepted in OEIS as A361098 in March 2023.

Updated 24 August 2024.