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_2(n) is the sum of the squares of the divisors of n (A001157).

Row sums of triangles A134575 and A134559. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 02 2007

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, 1964 (and various reprintings), p. 827.

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

D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; p. 11.

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].

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.

Index entries for "core" sequences

G.f.: Sum_{k>0} k^2 x^k/(1-x^k). Dirichlet g.f.: zeta(s)*zeta(s-2). - Michael Somos, Apr 05 2003

Multiplicative with a(p^e) = (p^(2e+2)-1)/(p^2-1). - David W. Wilson (davidwwilson(AT)comcast.net), Aug 01, 2001.

G.f. for sigma_k(n): Sum_{m>0} m^k*x^m/(1-x^m). - Vladeta Jovovic (vladeta(AT)eunet.rs), Oct 18 2002

Equals A127093 * [1, 2, 3,...]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), May 10 2007

Equals A051731 * [1, 4, 9, 16, 25,...]. A051731 * [1/1, 1/2, 1/3, 1/4,...] = [1/1, 5/4, 10/9, 21/16, 26/25,...]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 02 2007

Row sums of triangle A134841 - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 12 2007

with(numtheory); A001157 := n->sigma[2](n); [seq(sigma[2](n), n=1..100)];

Table[DivisorSigma[2, n], {n, 1, 50}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Mar 24 2006

(PARI) a(n)=if(n<1, 0, sigma(n, 2))

(PARI) a(n)=if(n<1, 0, direuler(p=2, n, 1/(1-X)/(1-p^2*X))[n])

(PARI) a(n)=if(n<1, 0, n*polcoeff(sum(k=1, n, x^k/(x^k-1)^2/k, x*O(x^n)), n)) /* Michael Somos Jan 29 2005 */

(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)); \\ A001157 sum of squares of divisors of n.

s=serconvol(s, c)

v=Vec(s)

(Other) sage: [sigma(n, 2)for n in xrange(1, 51)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 04 2009]

Cf. A000005, A000203, A001158, A001159.

Cf. A053807, A064602.

Cf. A127093.

Cf. A134841.

Sequence in context: A002791 A080399 A017667 this_sequence A002800 A132174 A132461

Adjacent sequences: A001154 A001155 A001156 this_sequence A001158 A001159 A001160

nonn,core,nice,easy,mult

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

More terms from Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Mar 24 2006