A013969 a(n) = sigma_21(n), the sum of the 21st powers of the divisors of n.

1, 2097153, 10460353204, 4398048608257, 476837158203126, 21936961102828212, 558545864083284008, 9223376434903384065, 109418989141972712413, 1000000476837160300278, 7400249944258160101212, 46005141850728850805428, 247064529073450392704414, 1171356134499851307229224

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, A013954-A013972, A017665-A017712.

Programs

Magma

[DivisorSigma(21,n): n in [1..50]]; // G. C. Greubel, Nov 03 2018

Mathematica

Table[DivisorSigma[21,n],{n,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 11 2009 *)

PARI

vector(50, n, sigma(n,21)) \\ G. C. Greubel, Nov 03 2018

Sage

[sigma(n,21)for n in range(1,13)] # Zerinvary Lajos, Jun 04 2009

Formula

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

Sum_{n>=1} a(n)/exp(2*Pi*n) = 77683/552 = Bernoulli(22)/44. - Vaclav Kotesovec, May 07 2023

From Amiram Eldar, Oct 29 2023: (Start)

Multiplicative with a(p^e) = (p^(21*e+21)-1)/(p^21-1).

Dirichlet g.f.: zeta(s)*zeta(s-21).

Sum_{k=1..n} a(k) = zeta(22) * n^22 / 22 + O(n^23). (End)