%I A001157 M3799 N1551
%S A001157 1,5,10,21,26,50,50,85,91,130,122,210,170,250,260,341,290,455,362,546,
%T A001157 500,610,530,850,651,850,820,1050,842,1300,962,1365,1220,1450,1300,1911,
%U A001157 1370,1810,1700,2210,1682,2500,1850,2562,2366,2650,2210,3410,2451,3255
%N A001157 sigma_2(n): sum of squares of divisors of n.
%C A001157 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).
%C A001157 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.
%C A001157 sigma_2(n) is the sum of the squares of the divisors of n (A001157).
%C A001157 Row sums of triangles A134575 and A134559. - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Nov 02 2007
%D A001157 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001157 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001157 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.
%D A001157 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 
               1976, page 38.
%D A001157 D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; p. 
               11.
%H A001157 T. D. Noe, <a href="b001157.txt">Table of n, a(n) for n = 1..10000</a>
%H A001157 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
               abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National 
               Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 
               [alternative scanned copy].
%H A001157 Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/
               index.html">Arithmetic and growth of periodic orbits</a>, J. Integer 
               Seqs., Vol. 4 (2001), #01.2.1.
%H A001157 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               DivisorFunction.html">Link to a section of The World of Mathematics.</
               a>
%H A001157 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A001157 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
%F A001157 Multiplicative with a(p^e) = (p^(2e+2)-1)/(p^2-1). - David W. Wilson 
               (davidwwilson(AT)comcast.net), Aug 01, 2001.
%F A001157 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
%F A001157 Equals A127093 * [1, 2, 3,...]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               May 10 2007
%F A001157 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
%F A001157 Row sums of triangle A134841 - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Nov 12 2007
%p A001157 with(numtheory); A001157 := n->sigma[2](n); [seq(sigma[2](n), n=1..100)];
%t A001157 Table[DivisorSigma[2, n], {n, 1, 50}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), 
               Mar 24 2006
%o A001157 (PARI) a(n)=if(n<1,0,sigma(n,2))
%o A001157 (PARI) a(n)=if(n<1,0,direuler(p=2,n,1/(1-X)/(1-p^2*X))[n])
%o A001157 (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 */
%o A001157 (PARI from Joerg Arndt (arndt(AT)jjj.de), May 03, 2008)
%o A001157 N=17; default(seriesprecision,N); x=z+O(z^(N+1))
%o A001157 c=sum(j=1,N,j*x^j); \\ log case
%o A001157 s=-log(prod(j=1,N,(1-x^j)^j)); \\ A001157 sum of squares of divisors 
               of n.
%o A001157 s=serconvol(s,c)
%o A001157 v=Vec(s)
%o A001157 (Other) sage: [sigma(n,2)for n in xrange(1,51)] # [From Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), Jun 04 2009]
%Y A001157 Cf. A000005, A000203, A001158, A001159.
%Y A001157 Cf. A053807, A064602.
%Y A001157 Cf. A127093.
%Y A001157 Cf. A134841.
%Y A001157 Sequence in context: A002791 A080399 A017667 this_sequence A002800 A132174 
               A132461
%Y A001157 Adjacent sequences: A001154 A001155 A001156 this_sequence A001158 A001159 
               A001160
%K A001157 nonn,core,nice,easy,mult
%O A001157 1,2
%A A001157 N. J. A. Sloane (njas(AT)research.att.com), R. K. Guy
%E A001157 More terms from Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), 
               Mar 24 2006