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

1,2

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

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Index entries for sequences related to sigma(n).

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)

Mathematica

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

Prog

(Sage) [sigma(n, 15)for n in range(1, 15)] # Zerinvary Lajos, Jun 04 2009
(Magma) [DivisorSigma(15, n): n in [1..20]]; // 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

Crossrefs

Cf. A000203, A001157-A001160, A013674, A013954-A013972, A017665-A017712.

Sequence in context: A303267 A323545 A017693 * A036093 A217357 A083603

Adjacent sequences: A013960 A013961 A013962 * A013964 A013965 A013966

Keyword

nonn,mult,easy

Author

N. J. A. Sloane

Status

approved