A013961 a(n) = sigma_13(n), the sum of the 13th powers of the divisors of n.

1, 8193, 1594324, 67117057, 1220703126, 13062296532, 96889010408, 549822930945, 2541867422653, 10001220711318, 34522712143932, 107006334784468, 302875106592254, 793811662272744, 1946196290656824, 4504149450301441, 9904578032905938, 20825519793796029, 42052983462257060

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^13*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003

Dirichlet g.f.: zeta(s-13)*zeta(s). - Ilya Gutkovskiy, Sep 10 2016

Empirical: Sum_{n>=1} a(n)/exp(2*Pi*n) = 1/24. - Simon Plouffe, Mar 01 2021

From Amiram Eldar, Oct 29 2023: (Start)

Multiplicative with a(p^e) = (p^(13*e+13)-1)/(p^13-1).

Sum_{k=1..n} a(k) = zeta(14) * n^14 / 14 + O(n^15). (End)

Maple

A013961 := proc(n)
    numtheory[sigma][13](n) ;
end proc: # R. J. Mathar, Sep 21 2017

Mathematica

DivisorSigma[13, Range[30]] (* Vincenzo Librandi, Sep 10 2016 *)

Prog

(Sage) [sigma(n, 13)for n in range(1, 16)] # Zerinvary Lajos, Jun 04 2009
(Magma) [DivisorSigma(13, 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^13*q^n/(1-q^n))) \\ Altug Alkan, Sep 10 2016
(PARI) a(n) = sigma(n, 13); \\ Michel Marcus, Sep 10 2016

Crossrefs

Cf. A000203, A001157-A001160, A013672, A013954-A013972, A017665-A017712.

Sequence in context: A323544 A230190 A017689 * A036091 A181134 A345635

Adjacent sequences: A013958 A013959 A013960 * A013962 A013963 A013964

Keyword

nonn,mult,easy

Author

N. J. A. Sloane

Status

approved