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A013957 sigma_9(n), the sum of the 9th powers of the divisors of n. 20
1, 513, 19684, 262657, 1953126, 10097892, 40353608, 134480385, 387440173, 1001953638, 2357947692, 5170140388, 10604499374, 20701400904, 38445332184, 68853957121, 118587876498, 198756808749, 322687697780, 513002215782 (list; graph; refs; listen; history; text; internal format)
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^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

MATHEMATICA

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

PROG

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

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

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

CROSSREFS

Sequence in context: A223651 A321565 A017681 * A294304 A036087 A007487

Adjacent sequences:  A013954 A013955 A013956 * A013958 A013959 A013960

KEYWORD

nonn,mult,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified December 14 11:52 EST 2018. Contains 318097 sequences. (Running on oeis4.)