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

1,2

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}.

Links

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

Example

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.

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]]]

Prog

(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

Crossrefs

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

Sequence in context: A209292 A185371 A071894 * A271876 A358533 A078444

Adjacent sequences: A301889 A301890 A301891 * A301893 A301894 A301895

Keyword

nonn

Author

Michael De Vlieger, Mar 28 2018

Status

approved