A013970 a(n) = sigma_22(n), the sum of the 22nd powers of the divisors of n.

1, 4194305, 31381059610, 17592190238721, 2384185791015626, 131621735227521050, 3909821048582988050, 73786993887028445185, 984770902214992292491, 10000002384185795209930, 81402749386839761113322, 552061570551763831158810, 3211838877954855105157370

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

Mathematica

lst={};Do[AppendTo[lst,DivisorSigma[22,n]],{n,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Mar 11 2009 *)
a[ n_] := DivisorSigma[ 22, n]; (* Michael Somos, Dec 19 2016 *)

PARI

vector(50, n, sigma(n,22)) \\ G. C. Greubel, Nov 03 2018

Sage

[sigma(n,22)for n in range(1,12)] # Zerinvary Lajos, Jun 04 2009

Formula

G.f.: Sum_{k>=1} k^22*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003

From Amiram Eldar, Oct 29 2023: (Start)

Multiplicative with a(p^e) = (p^(22*e+22)-1)/(p^22-1).

Dirichlet g.f.: zeta(s)*zeta(s-22).

Sum_{k=1..n} a(k) = zeta(23) * n^23 / 23 + O(n^24). (End)