A013959 a(n) = sigma_11(n), the sum of the 11th powers of the divisors of n.

1, 2049, 177148, 4196353, 48828126, 362976252, 1977326744, 8594130945, 31381236757, 100048830174, 285311670612, 743375541244, 1792160394038, 4051542498456, 8649804864648, 17600780175361, 34271896307634, 64300154115093, 116490258898220, 204900053024478

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

Related to congruence properties of the Ramanujan tau function since A000594(n) == a(n) (mod 691) = A046694(n). - Benoit Cloitre, Aug 28 2002

Links

• T. D. Noe, Table of n, a(n) for n=1..1000
• Index entries for sequences related to sigma(n).

Crossrefs

Cf. A000594, A027860, A046694.

Cf. A000203, A001157-A001160, A013670, A013954-A013972, A017665-A017712.

Programs

Magma

[DivisorSigma(11, n): n in [1..20]]; // Vincenzo Librandi, Sep 10 2016

Mathematica

Table[DivisorSigma[11, n], {n, 30}] (* Vincenzo Librandi, Sep 10 2016 *)

PARI

a(n)=sigma(n,11) \\ Charles R Greathouse IV, Apr 28 2011

PARI

my(N=99, q='q+O('q^N)); Vec(sum(n=1, N, n^11*q^n/(1-q^n))) \\ Altug Alkan, Sep 10 2016

Python

from sympy import divisor_sigma
def A013959(n): return divisor_sigma(n,11) # Chai Wah Wu, Nov 17 2022

Sage

[sigma(n,11)for n in range(1,18)] # Zerinvary Lajos, Jun 04 2009

Formula

G.f.: Sum_{k>=1} k^11*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003

Dirichlet g.f.: zeta(s-11)*zeta(s). - Ilya Gutkovskiy, Sep 10 2016

From Amiram Eldar, Oct 29 2023: (Start)

Multiplicative with a(p^e) = (p^(11*e+11)-1)/(p^11-1).

Sum_{k=1..n} a(k) = zeta(12) * n^12 / 12 + O(n^13). (End)