A013958 a(n) = sigma_10(n), the sum of the 10th powers of the divisors of n.

1, 1025, 59050, 1049601, 9765626, 60526250, 282475250, 1074791425, 3486843451, 10009766650, 25937424602, 61978939050, 137858491850, 289537131250, 576660215300, 1100586419201, 2015993900450, 3574014537275, 6131066257802, 10250010815226, 16680163512500, 26585860217050

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

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Index entries for sequences related to sigma(n).

Formula

G.f.: Sum_{k>=1} k^10*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003

L.g.f.: -log(Product_{k>=1} (1 - x^k)^(k^9)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 06 2017

From Amiram Eldar, Oct 29 2023: (Start)

Multiplicative with a(p^e) = (p^(10*e+10)-1)/(p^10-1).

Dirichlet g.f.: zeta(s)*zeta(s-10).

Sum_{k=1..n} a(k) = zeta(11) * n^11 / 11 + O(n^12). (End)

Mathematica

lst={}; Do[AppendTo[lst, DivisorSigma[10, n]], {n, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Mar 11 2009 *)
DivisorSigma[10, Range[20]] (* Harvey P. Dale, Jan 04 2012 *)

Prog

(Sage) [sigma(n, 10)for n in range(1, 18)] # Zerinvary Lajos, Jun 04 2009
(PARI) a(n)=sigma(n, 10) \\ Charles R Greathouse IV, Apr 28, 2011
(Magma) [DivisorSigma(10, n): n in [1..20]]; // Bruno Berselli, Apr 10 2013

Crossrefs

Cf. A000203, A001157-A001160, A013669, A013954-A013972, A017665-A017712.

Sequence in context: A321807 A351305 A017683 * A294305 A036088 A023002

Adjacent sequences: A013955 A013956 A013957 * A013959 A013960 A013961

Keyword

nonn,mult,easy

Author

N. J. A. Sloane

Status

approved