A013962 a(n) = sigma_14(n), the sum of the 14th powers of the divisors of n.

1, 16385, 4782970, 268451841, 6103515626, 78368963450, 678223072850, 4398314962945, 22876797237931, 100006103532010, 379749833583242, 1283997101947770, 3937376385699290, 11112685048647250, 29192932133689220, 72061992352890881, 168377826559400930

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• mult
• easy

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

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Index entries for sequences related to sigma(n).

Crossrefs

Cf. A000203, A001157-A001160, A013673, A013954-A013972, A017665-A017712.

Programs

Magma

[DivisorSigma(14, n): n in [1..20]]; // Vincenzo Librandi, Sep 10 2016

Mathematica

DivisorSigma[14,Range[20]] (* Harvey P. Dale, Mar 10 2013 *)

PARI

my(N=99, q='q+O('q^N)); Vec(sum(n=1, N, n^14*q^n/(1-q^n))) \\ Altug Alkan, Sep 10 2016

PARI

a(n) = sigma(n, 14); \\ Amiram Eldar, Oct 29 2023

Sage

[sigma(n,14) for n in range(1,16)] # Zerinvary Lajos, Jun 04 2009

Formula

G.f.: Sum_{k>=1} k^14*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003

Dirichlet g.f.: zeta(s-14)*zeta(s). - Ilya Gutkovskiy, Sep 10 2016

From Amiram Eldar, Oct 29 2023: (Start)

Multiplicative with a(p^e) = (p^(14*e+14)-1)/(p^14-1).

Sum_{k=1..n} a(k) = zeta(15) * n^15 / 15 + O(n^16). (End)