A013963 a(n) = sigma_15(n), the sum of the 15th powers of the divisors of n.

1, 32769, 14348908, 1073774593, 30517578126, 470199366252, 4747561509944, 35185445863425, 205891146443557, 1000030517610894, 4177248169415652, 15407492847694444, 51185893014090758, 155572843119354936, 437893920912786408, 1152956690052710401, 2862423051509815794

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, A013674, A013954-A013972, A017665-A017712.

Programs

Magma

[DivisorSigma(15, n): n in [1..20]]; // Vincenzo Librandi, Sep 10 2016

Mathematica

DivisorSigma[15, Range[30]] (* Vincenzo Librandi, Sep 10 2016 *)

PARI

my(N=99, q='q+O('q^N)); Vec(sum(n=1, N, n^15*q^n/(1-q^n))) \\ Altug Alkan, Sep 10 2016

PARI

a(n) = sigma(n, 15); \\ Amiram Eldar, Oct 29 2023

Sage

[sigma(n,15)for n in range(1,15)] # Zerinvary Lajos, Jun 04 2009

Formula

G.f.: Sum_{k>=1} k^15*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003

Dirichlet g.f.: zeta(s-15)*zeta(s). - Ilya Gutkovskiy, Sep 10 2016

From Amiram Eldar, Oct 29 2023: (Start)

Multiplicative with a(p^e) = (p^(15*e+15)-1)/(p^15-1).

Sum_{k=1..n} a(k) = zeta(16) * n^16 / 16 + O(n^17). (End)