1,2

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 sequencesA017665-A017712also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequencesA000203(k=1),A001157-A001160(k=2,3,4,5),A013954-A013972for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001

Vincenzo Librandi,Table of n, a(n) for n = 1..1000

Index entries for sequences related to sigma(n).

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

FromAmiram 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)

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

(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

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

Sequence in context:A342686A321809A017687*A036090A123094A226695

Adjacent sequences:A013957A013958A013959*A013961A013962A013963

nonn,mult,easy

approved