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.

sigma_3(n) is the sum of the cubes of the divisors of n (A001158).

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. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 38.

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.

Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Multiplicative with a(p^e) = (p^(3e+3)-1)/(p^3-1). - David W. Wilson (davidwwilson(AT)comcast.net), Aug 01, 2001.

G.f.: sum(k>=1, k^3*x^k/(1-x^k)). - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 21 2003

Equals A051731 * [1, 8, 27, 64, 125,...] = A127093 * [1, 4, 9, 16, 25,...]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 02 2007

Table[DivisorSigma[3, n], {n, 100}]

(PARI from Joerg Arndt (arndt(AT)jjj.de), May 03, 2008)

N=17; default(seriesprecision, N); x=z+O(z^(N+1))

c=sum(j=1, N, j*x^j); \\ log case

s=-log(prod(j=1, N, (1-x^j)^(j^2))); \\ A001158 sum of cubes of divisors of n.

s=serconvol(s, c)

v=Vec(s)

(Other) sage: [sigma(n, 3)for n in xrange(1, 40)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 04 2009]

Cf. A000005, A000203, A001157.

Cf. A051731, A127093.

Sequence in context: A062451 A065959 A017669 this_sequence A053819 A085292 A073706

Adjacent sequences: A001155 A001156 A001157 this_sequence A001159 A001160 A001161

nonn,easy,nice,mult

N. J. A. Sloane (njas(AT)research.att.com), R. K. Guy

Corrected Mathematica code T. D. Noe (noe(AT)sspectra.com), Mar 22 2009