This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A301892 #13 Jul 07 2019 05:51:56
%S A301892 1,2,3,5,8,11,14,15,26,36,44,49,58,76,131,156,174,206,266,308,339,388,
%T A301892 428,460,766,550,568,979,1124,1238,1411,1548,1659,1754,1983,2048,2160,
%U A301892 3689,4211,4617,5245,5731,6135,6482,7308,7539,7949,8477,9198,9681,10306
%N A301892 a(n) = A010846(A002182(n)).
%C A301892 We define an "n-regular" number as 1 <= m <= n such that m | n^e with integer e >= 0. The divisor d is a special case of regular number m such that d | n^e with e = 0 or e = 1. Regular numbers m can exceed n; we are concerned only with regulars m <= n herein.
%C A301892 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.
%C A301892 Only 13 HCNs less than 36 * 10^6 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.
%C A301892 Only 2 and 6 set records for the ratio A010846(n)/A000005(n).
%C A301892 Conjectures:
%C A301892 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:
%C A301892 {0, 1},
%C A301892 {{1,1}, {1,2}},
%C A301892 {{2,1}, {2,2}, {2,4}},
%C A301892 {{3,2}, {3,4}, {3,6}},
%C A301892 {{4,4}, {4,6}, {4,8}},
%C A301892 {5,12}.
%H A301892 Amiram Eldar, <a href="/A301892/b301892.txt">Table of n, a(n) for n = 1..135</a>
%e A301892 A002182(4) = 6. There are five numbers 1 <= m <= 6 such that m divides an integer power of 6: {1, 2, 3, 4, 6}. Thus, a(4) = 5.
%t A301892 With[{s = Array[DivisorSigma[0, #] &, 10^6]}, Map[With[{n = FirstPosition[s, #][[1]]}, Count[Range@ n, _?(PowerMod[n, Floor@ Log2@ n, #] == 0 &)]] &, Union@ FoldList[Max, s]]]
%o A301892 (PARI) a010846(n) = sum(k=1, n, if(gcd(n, k)-1, 0, moebius(k)*(n\k))) \\ after _Benoit Cloitre_ in A010846
%o A301892 r=0; for(x=1, oo, if(numdiv(x) > r, print1(a010846(x), ", "); r=numdiv(x))) \\ _Felix Fröhlich_, Mar 30 2018
%Y A301892 Cf. A000005, A002110, A002182, A010846, A060735, A244052.
%K A301892 nonn
%O A301892 1,2
%A A301892 _Michael De Vlieger_, Mar 28 2018