A301892 a(n) = A010846(A002182(n)).

1, 2, 3, 5, 8, 11, 14, 15, 26, 36, 44, 49, 58, 76, 131, 156, 174, 206, 266, 308, 339, 388, 428, 460, 766, 550, 568, 979, 1124, 1238, 1411, 1548, 1659, 1754, 1983, 2048, 2160, 3689, 4211, 4617, 5245, 5731, 6135, 6482, 7308, 7539, 7949, 8477, 9198, 9681, 10306

Offset: 1

Author

Michael De Vlieger, Mar 28 2018

Keywords

• nonn

Comments

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.

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 * 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.

Only 2 and 6 set records for the ratio A010846(n)/A000005(n).

Conjectures:

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}.

Examples

			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.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..135

Crossrefs

Cf. A000005, A002110, A002182, A010846, A060735, A244052.

Programs

Mathematica

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]]]

PARI

a010846(n) = sum(k=1, n, if(gcd(n, k)-1, 0, moebius(k)*(n\k))) \\ after Benoit Cloitre in A010846
r=0; for(x=1, oo, if(numdiv(x) > r, print1(a010846(x), ", "); r=numdiv(x))) \\ Felix Fröhlich, Mar 30 2018