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

1,2

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 (first 1000 terms from Harvey P. Dale)

Index entries for sequences related to sigma(n).

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)

Mathematica

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

Prog

(Sage) [sigma(n, 23)for n in range(1, 12)] # Zerinvary Lajos, Jun 04 2009
(PARI) vector(30, n, sigma(n, 23)) \\ G. C. Greubel, Nov 03 2018
(Magma) [DivisorSigma(23, n): n in [1..30]]; // G. C. Greubel, Nov 03 2018

Crossrefs

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

Sequence in context: A010811 A323660 A017709 * A036101 A283031 A160673

Adjacent sequences: A013968 A013969 A013970 * A013972 A013973 A013974

Keyword

nonn,easy,mult

Author

N. J. A. Sloane

Status

approved