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 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

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

From Amiram 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^21). (End)

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

(SageMath) [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: A010809 A323659 A017705 * A036099 A253058 A233979

Adjacent sequences: A013966 A013967 A013968 * A013970 A013971 A013972

nonn,easy,mult

approved