A013971 a(n) = sigma_23(n), the sum of the 23rd powers of the divisors of n.

1, 8388609, 94143178828, 70368752566273, 11920928955078126, 789730317205170252, 27368747340080916344, 590295880727458217985, 8862938119746644274757, 100000011920928963466734, 895430243255237372246532, 6624738056749922960468044, 41753905413413116367045798

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 (first 1000 terms from Harvey P. Dale)
• Index entries for sequences related to sigma(n).

Crossrefs

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

Programs

Magma

[DivisorSigma(23,n): n in [1..30]]; // G. C. Greubel, Nov 03 2018

Mathematica

DivisorSigma[23,Range[15]] (* Harvey P. Dale, May 02 2016 *)

PARI

vector(30, n, sigma(n,23)) \\ G. C. Greubel, Nov 03 2018

Sage

[sigma(n,23)for n in range(1,12)] # Zerinvary Lajos, Jun 04 2009

Formula

G.f.: Sum_{k>=1} k^23*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003

From Amiram Eldar, Oct 29 2023: (Start)

Multiplicative with a(p^e) = (p^(23*e+23)-1)/(p^23-1).

Dirichlet g.f.: zeta(s)*zeta(s-23).

Sum_{k=1..n} a(k) = zeta(24) * n^24 / 24 + O(n^25). (End)