A013972 a(n) = sigma_24(n), the sum of the 24th powers of the divisors of n.

1, 16777217, 282429536482, 281474993487873, 59604644775390626, 4738381620767930594, 191581231380566414402, 4722366764344638701569, 79766443077154939399843, 1000000059604644792167842, 9849732675807611094711842, 79496851942053939878082786, 542800770374370512771595362

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

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

Crossrefs

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

Programs

Magma

[DivisorSigma(24,n): n in [1..50]]; // G. C. Greubel, Nov 03 2018

Mathematica

Table[DivisorSigma[24,n],{n,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 11 2009 *)

PARI

a(n)=sigma(n,24) \\ Charles R Greathouse IV, Apr 28 2011

Sage

[sigma(n,24)for n in range(1,12)] # Zerinvary Lajos, Jun 04 2009

Formula

G.f.: Sum_{k>=1} k^24*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003

From Amiram Eldar, Oct 29 2023: (Start)

Multiplicative with a(p^e) = (p^(24*e+24)-1)/(p^24-1).

Dirichlet g.f.: zeta(s)*zeta(s-24).

Sum_{k=1..n} a(k) = zeta(25) * n^25 / 25 + O(n^26). (End)