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

1,2

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

Links

T. D. Noe, Table of n, a(n) for n=1..1000

Index entries for sequences related to sigma(n).

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

Cf. A000594, A027860, A046694.

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

Sequence in context: A230189 A321808 A017685 * A036089 A123095 A174752

Adjacent sequences: A013956 A013957 A013958 * A013960 A013961 A013962

Keyword

nonn,mult,easy

Author

N. J. A. Sloane

Status

approved