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 ), Apr 05 2001

Seiichi Manyama,Table of n, a(n) for n = 1..10000

Index entries for sequences related to sigma(n).

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

Sum_{n>=1} a(n)/exp(2*Pi*n) = 77683/552 = Bernoulli(22)/44. -Vaclav Kotesovec,May 07 2023

FromAmiram Eldar,Oct 29 2023: (Start)

Multiplicative with a(p^e) = (p^(21*e+21)-1)/(p^21-1).

Dirichlet g.f.: zeta(s)*zeta(s-21).

Sum_{k=1..n} a(k) = zeta(22) * n^22 / 22 + O(n^23). (End)

Table[DivisorSigma[21, n], {n, 50}] (*Vladimir Joseph Stephan Orlovsky,Mar 11 2009 *)

(Sage) [sigma(n, 21)for n in range(1, 13)] #Zerinvary Lajos,Jun 04 2009

(PARI) vector(50, n, sigma(n, 21)) \\G. C. Greubel,Nov 03 2018

(Magma) [DivisorSigma(21, n): n in [1..50]]; //G. C. Greubel,Nov 03 2018

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

Sequence in context:A010809A323659A017705*A036099A253058A233979

Adjacent sequences:A013966A013967A013968*A013970A013971A013972

nonn,easy,mult

approved