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
Related to congruence properties of the Ramanujan tau function since A000594(n) == a(n) (mod 691) = A046694(n). - Benoit Cloitre, Aug 28 2002
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)
MATHEMATICA
Table[DivisorSigma[11, n], {n, 30}] (* Vincenzo Librandi, Sep 10 2016 *)
PROG
(Sage) [sigma(n, 11)for n in range(1, 18)] # Zerinvary Lajos, Jun 04 2009
(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
(Magma) [DivisorSigma(11, n): n in [1..20]]; // Vincenzo Librandi, Sep 10 2016
(Python)
from sympy import divisor_sigma
def A013959(n): return divisor_sigma(n, 11) # Chai Wah Wu, Nov 17 2022
CROSSREFS
KEYWORD
nonn,mult,easy
AUTHOR
STATUS
approved