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

1,2

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).

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)

Mathematica

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

Prog

(Sage) [sigma(n, 12) for n in range(1, 17)] # Zerinvary Lajos, Jun 04 2009
(Magma) [DivisorSigma(12, n): n in [1..20]]; // Vincenzo Librandi, Sep 10 2016
(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

Crossrefs

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

Sequence in context: A342686 A321809 A017687 * A036090 A123094 A226695

Adjacent sequences: A013957 A013958 A013959 * A013961 A013962 A013963

Keyword

nonn,mult,easy

Author

N. J. A. Sloane

Status

approved