A013957 a(n) = sigma_9(n), the sum of the 9th powers of the divisors of n.

1, 513, 19684, 262657, 1953126, 10097892, 40353608, 134480385, 387440173, 1001953638, 2357947692, 5170140388, 10604499374, 20701400904, 38445332184, 68853957121, 118587876498, 198756808749, 322687697780, 513002215782, 794320419872, 1209627165996, 1801152661464

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• mult
• easy

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

Note that the sequence is not monotonically increasing, with a(4488) > a(4489) being the first of infinitely many examples. - Charles R Greathouse IV, Dec 28 2021

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• T. H. Grönwall, Some asymptotic expressions in the Theory of Numbers, Trans. Amer. Math. Soc., Vol. 14, No. 1 (1913), pp. 113-122.
• Index entries for sequences related to sigma(n).

Crossrefs

Cf. A000203, A001157-A001160, A013667, A013668, A013954-A013972, A017665-A017712.

Programs

Magma

[DivisorSigma(9,n): n in [1..20]]; // Bruno Berselli, Apr 10 2013

Mathematica

Table[DivisorSigma[9,n],{n,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 11 2009 *)

PARI

a(n)=if(n<1,0,sigma(n,9))

Sage

[sigma(n,9)for n in range(1,21)] # Zerinvary Lajos, Jun 04 2009

Formula

G.f.: Sum_{k>=1} k^9*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003

L.g.f.: -log(Product_{k>=1} (1 - x^k)^(k^8)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 06 2017

n^9 + 1 <= a(n) < zeta(9)*n^9. In particular, Grönwall proves lim sup a(n)/n^9 = zeta(9) = A013667. - Charles R Greathouse IV, Dec 27 2021

Sum_{n>=1} a(n)/exp(2*Pi*n) = 1/264 = Bernoulli(10)/20. - Vaclav Kotesovec, May 07 2023

From Amiram Eldar, Oct 29 2023: (Start)

Multiplicative with a(p^e) = (p^(9*e+9)-1)/(p^9-1).

Dirichlet g.f.: zeta(s)*zeta(s-9).

Sum_{k=1..n} a(k) = zeta(10) * n^10 / 10 + O(n^11). (End)