A013967 a(n) = sigma_19(n), the sum of the 19th powers of the divisors of n.

1, 524289, 1162261468, 274878431233, 19073486328126, 609360902796252, 11398895185373144, 144115462954287105, 1350851718835253557, 10000019073486852414, 61159090448414546292, 319480609006403630044, 1461920290375446110678, 5976315357844100294616

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• mult

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

• Seiichi Manyama, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to sigma(n).

Crossrefs

Cf. A000203, A001157-A001160, A013678, A013954-A013972, A017665-A017712.

Programs

Magma

[DivisorSigma(19,n): n in [1..50]]; // G. C. Greubel, Nov 03 2018

Mathematica

Table[DivisorSigma[19,n],{n,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 11 2009 *)

PARI

vector(50, n, sigma(n,19)) \\ G. C. Greubel, Nov 03 2018

Sage

[sigma(n,19)for n in range(1,13)] # Zerinvary Lajos, Jun 04 2009

Formula

G.f.: Sum_{k>=1} k^19*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003

From Amiram Eldar, Oct 29 2023: (Start)

Multiplicative with a(p^e) = (p^(19*e+19)-1)/(p^19-1).

Dirichlet g.f.: zeta(s)*zeta(s-19).

Sum_{k=1..n} a(k) = zeta(20) * n^20 / 20 + O(n^21). (End)