On the regular counting function applied to the highly composite numbers.

A301892(n) = A010846(A002182(n))
Michael De Vlieger, St. Louis, Missouri, 2018 0328 1030, revised 2018 0331 1415.

(This document is originally a text “data brief”. Data briefs intend to express observations and conjectures about sequences I’ve written. The data is the most important part of the document and can be very long. In this edition of the brief, the data is abbreviated, with links to the full analysis. Data briefs and the material in the math section of this site is intentionally starkly minimally designed so as to focus on the logic and data rather than formatting.)

We define an “n-regular” number as 1 ≤ m ≤ n such that m | ne with integer e ≥ 0. A number that is not regular is said to be “non-regular”.
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.
Regular numbers, apart from 1, occupy the cototient of n. The union of the regulars and the cototient leaves us with the “semitotative” for composite n > 6 listed in row n of A272619.
We are not concerned with “semidivisors” m such that m | ne with e > 1, listed in row n of A272618, but with the regular and divisor counting functions A010846(n) and A000005(n),
respectively.

Contents.

Table 1: Comparison of A010846(A002182(n)) and A002183(n).
Table 2: A301893(i) = Numbers k that set records for the ratio rcf(k)/τ(k).
Table 3: The intersection of A002182 and A244052.
(Observations and conjectures follow each table.)
Code.

Table 1: Comparison of A010846(A002182(n)) and A002183(n).

Questions:

  1. How many regular numbers do the highly composite numbers have, given that the divisor is a special case of regular number?
  2. Do they also set records for the regular counting function (i.e., are the highly composite numbers also "highly regular")?
  3. Are the HCNs representative of numbers that set records for the ratio rcf(k)/τ(k)?

(1) A002182(n)
(2) Index of A002182(n) in A244052. If A002182(n) is not in A244052, left blank.
(3) Index of A002182(n) in A301893 (see Table 2). (A301893 computed through 36 × 106).
(4) A301893(n) = numbers k that set records for the ratio rcf(k)/τ(k).
(5) A301892(n) = A010846(A002182(n)).
(6) A108602(n) = A001221(A002182(n)).
(7) A002183(n)/A301892(n)
(8) “Multiplicity Family” of A002182(n): MN( m/A002110(A108602(m)) ). (See Table 4).
MN = Multiplicity Notation = A054841(a(n)) here represents the reverse of A054841 in OEIS; i.e., “little-endian” rather than big-endian, e.g., A054841(84) = 1013 but here it is 2101. We use A054841 or “multiplicity notation” here since the terms of a(n) are products of relatively small primes and the notation succinctly expresses their prime decomposition. This is notation Achim Flammenkamp employed in his studies of highly composite numbers.

n (1) (2) (3) (4) (5) (6) (7) (8)
--- -------- ----- ----- ----- ----- ---------- ---- -------
1 1 1 1 1 1 1 0 0
2 2 2 2 2 1 1 0
3 4 3 3 3 1 1 1
4 6 4 2 4 5 0.8 2 0
5 12 6 6 8 0.75 2 1
6 24 8 8 11 0.727273 2 2
7 36 9 14 0.642857 2 11
8 48 10 15 0.666667 2 3
9 60 11 12 26 0.461538 3 1
10 120 14 16 36 0.444444 3 2
11 180 16 18 44 0.409091 3 11
12 240 20 49 0.408163 3 3
13 360 24 58 0.413793 3 21
14 720 30 76 0.394737 3 31
15 840 22 32 131 0.244275 4 2
16 1260 24 36 156 0.230769 4 11
17 1680 26 40 174 0.229885 4 3
18 2520 48 206 0.23301 4 21
19 5040 60 266 0.225564 4 31
20 7560 64 308 0.207792 4 22
21 10080 72 339 0.212389 4 41
22 15120 80 388 0.206186 4 32
23 20160 84 428 0.196262 4 51
24 25200 90 460 0.195652 4 311
25 27720 47 96 766 0.125326 5 21
26 45360 100 550 0.181818 4 33
27 50400 108 568 0.190141 4 411
28 55440 120 979 0.122574 5 31
29 83160 128 1124 0.113879 5 22
30 110880 144 1238 0.116317 5 41
31 166320 160 1411 0.113395 5 32
32 221760 168 1548 0.108527 5 51
33 277200 180 1659 0.108499 5 311
34 332640 192 1754 0.109464 5 42
35 498960 200 1983 0.100857 5 33
36 554400 216 2048 0.105469 5 411
37 665280 224 2160 0.103704 5 52
38 720720 240 3689 0.065058 6 31
39 1081080 256 4211 0.060793 6 22
40 1441440 288 4617 0.062378 6 41
41 2162160 320 5245 0.061011 6 32
42 2882880 336 5731 0.058629 6 51
43 3603600 360 6135 0.05868 6 311
44 4324320 384 6482 0.059241 6 42
45 6486480 400 7308 0.054735 6 33
46 7207200 432 7539 0.057302 6 411
47 8648640 448 7949 0.056359 6 52
48 10810800 480 8477 0.056624 6 321
49 14414400 504 9198 0.054795 6 511
50 17297280 512 9681 0.052887 6 62
51 21621600 576 10306 0.05589 6 421
52 32432400 600 11515 0.052106 6 331
53 36756720 - 640 19994 0.03201 7 32
54 43243200 - 672 12448 0.053985 6 521
55 61261200 - 720 23259 0.030956 7 311
56 73513440 - 768 24532 0.031306 7 42
57 110270160 - 800 27570 0.029017 7 33
58 122522400 - 864 28408 0.030414 7 411
59 147026880 - 896 29917 0.02995 7 52
60 183783600 - 960 31849 0.030142 7 321
61 245044800 - 1008 34500 0.029217 7 511
62 294053760 - 1024 36278 0.028227 7 62
63 367567200 - 1152 38557 0.029878 7 421
64 551350800 - 1200 43016 0.027897 7 331
65 698377680 - 1280 77028 0.016617 8 32
66 735134400 - 1344 46436 0.028943 7 521
67 1102701600 - 1440 51652 0.027879 7 431
68 1396755360 - 1536 93907 0.016357 8 42
69 2095133040 - 1600 105210 0.015208 8 33
70 2205403200 - 1680 61730 0.027215 7 531
71 2327925600 - 1728 108330 0.015951 8 411
72 2793510720 - 1792 113928 0.015729 8 52
73 3491888400 - 1920 121118 0.015852 8 321
74 4655851200 - 2016 130970 0.015393 8 511
75 5587021440 - 2048 137566 0.014887 8 62
76 6983776800 - 2304 146038 0.015777 8 421
77 10475665200 - 2400 162599 0.01476 8 331
78 13967553600 - 2688 175323 0.015332 8 521
79 20951330400 - 2880 194715 0.014791 8 431
80 27935107200 - 3072 209588 0.014657 8 621
81 41902660800 - 3360 232236 0.014468 8 531
82 48886437600 - 3456 241387 0.014317 8 4211
83 64250746560 - 3584 440351 0.008139 9 52
84 73329656400 - 3600 266983 0.013484 8 3311
85 80313433200 - 3840 467403 0.008216 9 321
86 97772875200 - 4032 286547 0.014071 8 5211
87 128501493120 - 4096 529244 0.007739 9 62
88 146659312800 - 4320 316236 0.013661 8 4311
89 160626866400 - 4608 561065 0.008213 9 421
90 240940299600 - 4800 623240 0.007702 9 331
91 293318625600 - 5040 373244 0.013503 8 5311
92 321253732800 - 5376 670993 0.008012 9 521
93 481880599200 - 5760 743774 0.007744 9 431
94 642507465600 - 6144 799574 0.007684 9 621
95 963761198400 - 6720 884524 0.007597 9 531
96 1124388064800 - 6912 918857 0.007522 9 4211
97 1606268664000 - 7168 1002878 0.007147 9 522
98 1686582097200 - 7200 1014877 0.007094 9 3311
99 1927522396800 - 7680 1048383 0.007326 9 631
100 2248776129600 - 8064 1088291 0.00741 9 5211
101 3212537328000 - 8192 1185830 0.006908 9 622
102 3373164194400 - 8640 1199750 0.007202 9 4311
103 4497552259200 - 9216 1284854 0.007173 9 6211
104 6746328388800 - 10080 1413882 0.007129 9 5311
105 8995104518400 - 10368 1512270 0.006856 9 7211
106 9316358251200 - 10752 2607576 0.004123 10 521
107 13492656777600 - 11520 1661238 0.006935 9 6311
108 18632716502400 - 12288 3095217 0.00397 10 621
109 26985313555200 - 12960 1946228 0.006659 9 7311
110 27949074753600 - 13440 3417004 0.003933 10 531
111 32607253879200 - 13824 3547008 0.003897 10 4211
112 46581791256000 - 14336 3864992 0.003709 10 522
113 48910880818800 - 14400 3910396 0.003682 10 3311
114 55898149507200 - 15360 4037208 0.003805 10 631
115 65214507758400 - 16128 4188170 0.003851 10 5211
116 93163582512000 - 16384 4557034 0.003595 10 622
117 97821761637600 - 17280 4609691 0.003749 10 4311
118 130429015516800 - 18432 4931474 0.003738 10 6211
119 195643523275200 - 20160 5419232 0.00372 10 5311
120 260858031033600 - 20736 5791113 0.003581 10 7211
121 288807105787200 - 21504 10174493 0.002114 11 521
122 391287046550400 - 23040 6354166 0.003626 10 6311
123 577614211574400 - 24576 12036043 0.002042 11 621
124 782574093100800 - 25920 7431427 0.003488 10 7311
125 866421317361600 - 26880 13263098 0.002027 11 531
126 1010824870255200 - 27648 13758600 0.00201 11 4211
127 1444035528936000 - 28672 14970020 0.001915 11 522
128 1516237305382800 - 28800 15143047 0.001902 11 3311
129 1732842634723200 - 30720 15625881 0.001966 11 631
130 2021649740510400 - 32256 16200656 0.001991 11 5211
131 2888071057872000 - 32768 17604683 0.001861 11 622
132 3032474610765600 - 34560 17805134 0.001941 11 4311
133 4043299481020800 - 36864 19029413 0.001937 11 6211
134 6064949221531200 - 40320 20884747 0.001931 11 5311
135 8086598962041600 - 41472 22298971 0.00186 11 7211

Observations

(These observations do not include those well-known by those interested in HCNs)

  1. The ratio τ(x)/rcf(x) for x = A002182(n) generally declines as x increases. This implies that for numbers that set records in the divisor counting function, the number of regular numbers is very much higher as n increases.
  2. Only 13 HCNs are also “highly regular”, i.e., appear in A244052. The largest HCN that is also highly regular is 27720, the 25th HCN and the 47th highly regular number. This seems to be associated with the fact that the prime decomposition of numbers in A002182 involves “more enriched” small primes as n increases, while the numbers in A244052 are either squarefree or in A060735.
  3. Only 2 and 6 set records for the ratio τ(x)/rcf(x). In other words, the highly composite numbers seem generally not representative of the highest possible ratio of number of divisors to the number of regulars.
  4. The value of A301892(n) jumps when ω(n) increases, in line with what we would expect of the regular counting function regarding not-necessarily squarefree products of n versus (n + 1) smallest primes.

Conjectures

  1. 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}.
    A "tier" n consists of all terms m in A244052 with A002110(n) ≤ m < A002110(n + 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}
    (It is this observation that spurred development of Tables 3, 4, 6, and Graph 5.)
  2. If it is true that the only non-squarefree terms in A301893(n) are {18, 150}, then
    the intersection of A002182 and A301893(n) is finite and consists of {1, 6}. (see Table 2 Conjecture 1).

Table 2: A301893(i) = Numbers k that set records for the ratio rcf(k)/τ(k).

Question:

  1. What numbers set records for the ratio of the regular counting function and the divisor counting function?

(1) k = A301893(i)
(2) Index of A301893(i) in A244052. If A301893(i) is not in A244052, left blank.
(3) rcf(k) = A010846(k).
(4) τ(k).
(5) τ(k)/rcf(k)
PC(n) = A287352(n) is the "pi-code" or first differences of indices of prime divisors p of n, e.g., A287352(60) = 1,0,1,1 since 60 = 2 × 2 × 3 × 5. The terms are delimited by (.).

i k = (1) (2) (3) (4) (5) PC(k)
--- -------- ----- ----- ----- ---------- ----------
1 1 1 1 1 1 0
2 6 4 5 4 0.8 1.1
3 10 5 6 4 0.666667 1.2
4 18 7 10 6 0.6 1.1.0          
5 22     7 4 0.571429 1.4
6 30 9 18 8 0.444444 1.1.1          
7 42 10 19 8 0.421053 1.1.2          
8 66     22 8 0.363636 1.1.3          
9 78     23 8 0.347826 1.1.4          
10 102     25 8 0.32 1.1.5          
11 114     26 8 0.307692 1.1.6          
12 138     27 8 0.296296 1.1.7          
13 150 15 41 12 0.292683 1.1.1.0        
14 174     29 8 0.275862 1.1.8          
15 210 17 68 16 0.235294 1.1.1.1        
16 330 18 77 16 0.207792 1.1.1.2        
17 390 19 80 16 0.2 1.1.1.3        
18 510     86 16 0.186047 1.1.1.4        
19 570     88 16 0.181818 1.1.1.5        
20 690     94 16 0.170213 1.1.1.6        
21 870     101 16 0.158416 1.1.1.7        
22 1110     106 16 0.150943 1.1.1.9        
23 1230     110 16 0.145455 1.1.1.10       
24 1290     112 16 0.142857 1.1.1.11       
25 1410     114 16 0.140351 1.1.1.12       
26 1590     118 16 0.135593 1.1.1.13       
27 1770     121 16 0.132231 1.1.1.14       
28 1830     122 16 0.131148 1.1.1.15       
29 2010     126 16 0.126984 1.1.1.16       
30 2130     128 16 0.125 1.1.1.17       
31 2190     130 16 0.123077 1.1.1.18       
32 2310 29 283 32 0.113074 1.1.1.1.1      
33 2730 30 295 32 0.108475 1.1.1.1.2      
34 3570 31 313 32 0.102236 1.1.1.1.3      
35 3990 32 322 32 0.099379 1.1.1.1.4      
36 4830     339 32 0.094395 1.1.1.1.5      

(See full table of 1114 rows in this text file.)

Observations

  1. There are 2 non-squarefree terms {18, 150} less than 36 × 106.
  2. No primes p appear in A301893. This is because all regular m divide p, and since all the regulars of 1 also divide 1, no primes appear in A301893. If we were to ignore 1, then the prime 2 would set a record for the ratio of rcf(k)/τ(k).
  3. The values of τ(A301893(i)) are in A000079, i.e., powers 2e except e = 1 (see number 2 above).
  4. Aside from the 2 non-squarefree terms, many terms m are products of A002110(j) × ph, with h > j 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) × ph with 2 ≤ h ≤ 8.
  5. There are a few terms of the form A002110(j) × pg × ph, with h > g > j + 1. 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.
  6. Regarding A001221(A301893(i)), the sequence seems to be divided into “tiers” wherein ω(A301893(i)) either increases or holds as i increases. Thus we might speak of “tiers” for all terms m ≤ A002110(6) = 30030. However, 36330 immediately follows 30030 in the sequence and ω(36330) = 5. For m > 30030, the numbers m with omega of a particular value are increasingly mixed with terms that have a different value. These intermingled “tiers” do seem to have a floor and ceiling. The tiers consist of terms as described in number 3 above:
    • Tier 0 contains {1}.
    • There is no tier 1 since this would contain numbers m with ω(m) = 1.
    • Tier 2 contains {6, 10, 18, 22}; 12 is missing since rcf(12)/τ(12) = 8/6 was bested by 10 with ratio 6/4. Otherwise, A002110(1) × h with 1 ≤ h ≤ 4 and h ≠ 3, also excepting the term 18.
    • Tier 3 contains A002110(2) × h with 1 ≤ h ≤ 8, as well as 150 = 2 × 3 × 5².
    • Tier 4 contains A002110(3) × h with 1 ≤ h ≤ 18, and h ≠ 8.
    • Tier 5 contains A002110(4) × h with 1 ≤ h ≤ 30. This tier is interrupted by A002110(6) as before mentioned, but continues after 30030 with 36 ≤ h ≤ 38. There are further intrusions of numbers with ω(m) = 6, as well as missing values in the range 47 ≤ h ≤ 356. The 5th tier seems to terminate at A301893(319) = A002110(7), whereinafter no number with ω(5) appears.
    • “Tier” 6 has a floor at A002110(6) and its ceiling has not yet been seen in computation.
  7. “Messy bands” of terms m in A244052 appear in A301893. The indices j of these terms m in A244052 appear as follows, usually with significant intrusions of A301893 not in A244052:
    • A244052 tier 0: j = 1 at i = 1.
    • A244052 tier 2: j = {4, 5, 7} starting at A301893(3) = 6 = A002110(2); includes the non-squarefree 18.
    • A244052 tier 3: j = {9, 10, 15} starting at A301893(6) = 30 = A002110(3); includes the non-squarefree 150.
    • A244052 tier 4: 17 ≤ j ≤ 19 at 15 ≤ i ≤ 17 respectively. (A301893(15) = 210 = A002110(4)).
    • A244052 tier 5: 29 ≤ j ≤ 32, at 32 ≤ i ≤ 35 respectively. (A301893(32) = 2310 = A002110(5)).
    • A244052 tier 6: 48 ≤ j ≤ 53 starting at A301893(62) = 30030 = A002110(6).
    • A244052 tier 7: 72 ≤ j ≤ 77 starting at A301893(319) = 510510 = A002110(7).
    • A244052 tier 7: 104 ≤ j ≤ 111* starting at A301893(777) = 9699690 = A002110(8).

    8. * it is unclear whether this is the maximum value of j for m in A244052 tier 7.

Conjectures

  1. The only non-squarefree terms in A301893 are 18 and 150.
  2. Primorials A002110(j) for j = 0 and j > 2 appear in A301893.
  3. Aside from 18 and 150, there are a few of the smallest terms k in tier t of A244052 in A301893. These terms are squarefree and also appear in A288813.
  4. There will always be “intrusions” of m of ω(m) < t among terms k in tier t of A244052 with t > 6, equating to m ≥ A002110(6) i.e., m ≥ 30030. Corollary to this is that the ranges j = 1, 17 ≤ j ≤ 19 and 29 ≤ j ≤ 32 in A244052 are the only contiguous terms of A244052 tier t in A301893.

Table 3: The intersection of A002182 and A244052.

Based on Conjecture 1.1, we re-examine the terms m that are in both A002182 and A244052, then produce a chart plotting them with n = primorial A002110(ω(m)) and k = m/A002110(ω(m)).

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}.
A “tier” n consists of all terms m in A244052 with A002110(n) ≤ m < A002110(n + 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 plotted below. The top figure is the coordinates (n,k) in A060735. The middle figure is the decimal number, and the bottom set of numbers delimited by “.” are the exponents of the prime pertaining to the place in which the exponent appears. For example, “2.1.1” → 2² × 3 × 5 = 60.

 

Observations

  1. The terms in the multiplicity notations of these numbers hold or decrease. For k = 3 or k = 5, we would have 1.2, 1.2.1, 1.2.1.1, or 1.1.2, 1.1.2.1, 1.1.2.1.1, etc. Such configurations of prime divisors are absent from terms in A002182.
  2. If we were to extend the irregular triangle beyond k < p(n + 1), then we could perhaps plot all terms of A002182 on a grid. In the grid below, dots “.” appear for terms in A060735, if not occupied by a term in A002182, which if in A060735 is followed by an asterisk. These dotted and asterisked positions are fully occupied by many but not all terms in A244052 and illustrates some of the “divergence” of the two record-setter sequences.

  1. Since we observe the terms plotted on a grid tend to occupy certain values of k, we might condense the grid and merely annotate coordinates for each of the terms. We can use
    {ω(m),m/A002110(ω(m))} = {A108602(i), A002182(i) / A002110(A108602(i))} to furnish T(n,k).

Continuing investigation.

Note: the consideration of Table 3 has led to further investigation that continues at this data brief.

If we take the investigation quite far, we can produce the following graph of {m/A002110(ω(m)), ω(m)}, eliminating columns m/A002110(ω(m)) without HCNs. The black pixels represent an HCN (i.e., m in A002182) while the red pixels represent an SHCN (i.e., m in A002201, also in A002182, since A002201 is a subset of A002182). The HCNs plotted here derive from [1], while the SHCNs plotted in red derive from my processing the b-file at A000705.

Concerns sequences:

A000005: Divisor counting function τ(n), i.e., number of numbers 1 ≤ d ≤ n such that d | ne with 0 ≤ e ≤ 1.
A000079: Nonnegative integer powers of 2.
A001221: ω(n) = Number of distinct prime divisors of n.
A002110: The primorials.
A002182: Highly composite numbers, i.e., where records are set in A000005.
A002183: Records in A000005 = A000005(A002182(n)).
A010846: Regular counting function, abbreviated rcf(n), i.e., number of numbers 1 ≤ m ≤ n such that m | ne with e ≥ 0.
A054841: For n = Product_pe, write ek in the k-th place, or write 0 if pk does not divide n.
A060735: Irregular triangle read by rows: T(n, k) = k × A002110(n) for 1 ≤ k < prime(n + 1).
A108602: A001221(A002182(n)).
A244052: Highly regular numbers, i.e., where records are set in A010846.
A288813: 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.
A301892: a(n) = A010846(A002182(n)).
A301893: Where records occur in A010846(n)/A000005(n).
See [2] for these and other A-number sequences in this work.

References

[1]: Achim Flammenkamp, “Highly Composite Numbers”, retrieved 2018 0329 2330 GMT. Downloads appear at the bottom of page and require “gunzip” application.
[2]: Neil Sloane, The Online Encyclopedia of Integer Sequences, retrieved 2018 0329 2330 GMT. See “Concerns Sequences” for individual links.

Original text-format data brief can be seen here.

////// Revision Record //////

201803281015 Created.
201803281330 A301892 extended to 115 terms.
201803281945 A301892 extended to 125 terms.
201803282030 Table 3 added.
201803290830 A301892 extended to 135 terms.
201803290845 Converted Table 1 last column to MN( A002182(n)/A002110(A108602(n)) ).
201803310845 Data brief converted to HTML; portion ascribed to A301413 partitioned.
201803311415 Improved tables and charts.

(Updated 29 March 2018)