A013956 a(n) = sigma_8(n), the sum of the 8th powers of the divisors of n.

1, 257, 6562, 65793, 390626, 1686434, 5764802, 16843009, 43053283, 100390882, 214358882, 431733666, 815730722, 1481554114, 2563287812, 4311810305, 6975757442, 11064693731, 16983563042, 25700456418, 37828630724, 55090232674, 78310985282, 110523825058, 152588281251

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

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, A013667, A013954-A013972, A017665-A017712.

Programs

Magma

[DivisorSigma(8,n): n in [1..30]]; // Bruno Berselli, Apr 10 2013

Mathematica

Table[DivisorSigma[8,n],{n,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 11 2009 *)

PARI

a(n)=sigma(n,8) \\ Charles R Greathouse IV, Apr 28 2011

Sage

[sigma(n,8)for n in range(1,21)] # Zerinvary Lajos, Jun 04 2009

Formula

G.f.: Sum_{k>=1} k^8*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003

L.g.f.: -log(Product_{k>=1} (1 - x^k)^(k^7)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 06 2017

From Amiram Eldar, Oct 29 2023: (Start)

Multiplicative with a(p^e) = (p^(8*e+8)-1)/(p^8-1).

Dirichlet g.f.: zeta(s)*zeta(s-8).

Sum_{k=1..n} a(k) = zeta(9) * n^9 / 9 + O(n^10). (End)