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

1,2

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

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Index entries for sequences related to sigma(n).

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)

Mathematica

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

Prog

(Sage) [sigma(n, 19)for n in range(1, 13)] # Zerinvary Lajos, Jun 04 2009
(PARI) vector(50, n, sigma(n, 19)) \\ G. C. Greubel, Nov 03 2018
(Magma) [DivisorSigma(19, n): n in [1..50]]; // G. C. Greubel, Nov 03 2018

Crossrefs

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

Sequence in context: A236227 A320345 A017701 * A036097 A254039 A170802

Adjacent sequences: A013964 A013965 A013966 * A013968 A013969 A013970

Keyword

nonn,easy,mult

Author

N. J. A. Sloane

Status

approved