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

1,2

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

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

Index entries for sequences related to sigma(n).

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)

Mathematica

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

Prog

(Sage) [sigma(n, 20)for n in range(1, 13)] # Zerinvary Lajos, Jun 04 2009
(PARI) vector(50, n, sigma(n, 20)) \\ G. C. Greubel, Nov 03 2018
(Magma) [DivisorSigma(20, n): n in [1..50]]; // G. C. Greubel, Nov 03 2018

Crossrefs

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

Sequence in context: A370255 A351316 A017703 * A036098 A203668 A253312

Adjacent sequences: A013965 A013966 A013967 * A013969 A013970 A013971

Keyword

nonn,easy,mult

Author

N. J. A. Sloane

Status

approved