A013965 a(n) = sigma_17(n), the sum of the 17th powers of the divisors of n.

1, 131073, 129140164, 17180000257, 762939453126, 16926788715972, 232630513987208, 2251816993685505, 16677181828806733, 100000762939584198, 505447028499293772, 2218628050709022148, 8650415919381337934, 30491579359845314184, 98526126098761952664

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

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

Crossrefs

Cf. A000203, A001157-A001160, A013676, A013954-A013972, A017665-A017712.

Programs

Magma

[DivisorSigma(17, n): n in [1..20]]; // Vincenzo Librandi, Sep 10 2016

Mathematica

DivisorSigma[17,Range[20]] (* Harvey P. Dale, May 30 2013 *)

PARI

my(N=99, q='q+O('q^N)); Vec(sum(n=1, N, n^17*q^n/(1-q^n))) \\ Altug Alkan, Sep 10 2016

PARI

a(n) = sigma(n, 17); \\ Amiram Eldar, Oct 29 2023

Sage

[sigma(n,17)for n in range(1,14)] # Zerinvary Lajos, Jun 04 2009

Formula

G.f.: Sum_{k>=1} k^17*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003

Dirichlet g.f.: zeta(s-17)*zeta(s). - Ilya Gutkovskiy, Sep 10 2016

Empirical: Sum_{n>=1} a(n)/exp(2*Pi*n) = 43867/28728, also equals Bernoulli(18)/36. - Simon Plouffe, May 06 2023

From Amiram Eldar, Oct 29 2023: (Start)

Multiplicative with a(p^e) = (p^(17*e+17)-1)/(p^17-1).

Sum_{k=1..n} a(k) = zeta(18) * n^18 / 18 + O(n^19). (End)