1,2

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. - comment from Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 827.

T. D. Noe, Table of n, a(n) for n = 1..10000

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 827.

Multiplicative with a(p^e) = (p^(5e+5)-1)/(p^5-1). - David W. Wilson (davidwwilson(AT)comcast.net), Aug 01, 2001.

G.f. sum(k>=1, k^5*x^k/(1-x^k)). - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 21 2003

lst={}; Do[AppendTo[lst, DivisorSigma[5, n]], {n, 5!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Mar 11 2009]

(Other) sage: [sigma(n, 5)for n in xrange(1, 30)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 04 2009]

Cf. A000005, A000203, A001157-A001159.

Sequence in context: A088703 A034679 A017673 this_sequence A061223 A119782 A008515

Adjacent sequences: A001157 A001158 A001159 this_sequence A001161 A001162 A001163

nonn,easy,mult

N. J. A. Sloane (njas(AT)research.att.com).