A013968 a(n) = sigma_20(n), the sum of the 20th powers of the divisors of n.

1, 1048577, 3486784402, 1099512676353, 95367431640626, 3656161927895954, 79792266297612002, 1152922604119523329, 12157665462543713203, 100000095367432689202, 672749994932560009202, 3833763649708914645906, 19004963774880799438802, 83668335217551100221154

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• mult

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

• Seiichi Manyama, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to sigma(n).

Crossrefs

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

Programs

Magma

[DivisorSigma(20,n): n in [1..50]]; // G. C. Greubel, Nov 03 2018

Mathematica

DivisorSigma[20,Range[20]] (* Harvey P. Dale, Jul 26 2015 *)

PARI

vector(50, n, sigma(n,20)) \\ G. C. Greubel, Nov 03 2018

Sage

[sigma(n,20)for n in range(1,13)] # Zerinvary Lajos, Jun 04 2009

Formula

G.f.: Sum_{k>=1} k^20*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003

From Amiram Eldar, Oct 29 2023: (Start)

Multiplicative with a(p^e) = (p^(20*e+20)-1)/(p^20-1).

Dirichlet g.f.: zeta(s)*zeta(s-20).

Sum_{k=1..n} a(k) = zeta(21) * n^21 / 21 + O(n^22). (End)