A013960 a(n) = sigma_12(n), the sum of the 12th powers of the divisors of n.

1, 4097, 531442, 16781313, 244140626, 2177317874, 13841287202, 68736258049, 282430067923, 1000244144722, 3138428376722, 8918294543346, 23298085122482, 56707753666594, 129746582562692, 281543712968705, 582622237229762, 1157115988280531, 2213314919066162

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• mult
• easy

Comments

If the canonical factorization of n into prime powers is the product of p^e(p) then sigma_k(n) = Product_p ((p^((e(p)+1)*k))-1)/(p^k-1).

Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Index entries for sequences related to sigma(n).

Crossrefs

Cf. A000203, A001157-A001160, A013671, A013954-A013972, A017665-A017712.

Programs

Magma

[DivisorSigma(12, n): n in [1..20]]; // Vincenzo Librandi, Sep 10 2016

Mathematica

DivisorSigma[12,Range[20]] (* Harvey P. Dale, Jan 28 2015 *)

PARI

my(N=99, q='q+O('q^N)); Vec(sum(n=1, N, n^12*q^n/(1-q^n))) \\ Altug Alkan, Sep 10 2016

PARI

a(n) = sigma(n, 12); \\ Amiram Eldar, Oct 29 2023

Sage

[sigma(n,12) for n in range(1,17)] # Zerinvary Lajos, Jun 04 2009

Formula

G.f.: Sum_{k>=1} k^12*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003

Dirichlet g.f.: zeta(s-12)*zeta(s). - Ilya Gutkovskiy, Sep 10 2016

From Amiram Eldar, Oct 29 2023: (Start)

Multiplicative with a(p^e) = (p^(12*e+12)-1)/(p^12-1).

Sum_{k=1..n} a(k) = zeta(13) * n^13 / 13 + O(n^14). (End)