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A013967 a(n) = sigma_19(n), the sum of the 19th powers of the divisors of n. 8
1, 524289, 1162261468, 274878431233, 19073486328126, 609360902796252, 11398895185373144, 144115462954287105, 1350851718835253557, 10000019073486852414, 61159090448414546292, 319480609006403630044 (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

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Index entries for sequences related to sigma(n)

FORMULA

G.f.: Sum_{k>=1} k^19*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003

MATHEMATICA

Table[DivisorSigma[19, n], {n, 50}] (* Vladimir Joseph Stephan Orlovsky, Mar 11 2009 *)

PROG

(Sage) [sigma(n, 19)for n in range(1, 13)] # Zerinvary Lajos, Jun 04 2009

(PARI) vector(50, n, sigma(n, 19)) \\ G. C. Greubel, Nov 03 2018

(MAGMA) [DivisorSigma(19, n): n in [1..50]]; // G. C. Greubel, Nov 03 2018

CROSSREFS

Sequence in context: A236227 A320345 A017701 * A036097 A254039 A170802

Adjacent sequences:  A013964 A013965 A013966 * A013968 A013969 A013970

KEYWORD

nonn,mult

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified April 6 08:53 EDT 2020. Contains 333268 sequences. (Running on oeis4.)