Search: id:A001158
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%I A001158 M4605 N1964
%S A001158 1,9,28,73,126,252,344,585,757,1134,1332,2044,2198,3096,3528,4681,
%T A001158 4914,6813,6860,9198,9632,11988,12168,16380,15751,19782,20440,25112,
%U A001158 24390,31752,29792,37449,37296,44226,43344,55261,50654,61740,61544
%N A001158 sigma_3(n): sum of cubes of divisors of n.
%C A001158 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 A001158 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 A001158 sigma_3(n) is the sum of the cubes of the divisors of n (A001158).
%D A001158 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001158 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001158 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, 
               National Bureau of Standards Applied Math.Series 55, Tenth Printing, 
               1972, p. 827.
%D A001158 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 
               1976, page 38.
%H A001158 T. D. Noe, <a href="b001158.txt">Table of n, a(n) for n = 1..10000</a>
%H A001158 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 A001158 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/
               Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</
               a>, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 
               1972, p. 827.
%H A001158 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 A001158 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               DivisorFunction.html">Link to a section of The World of Mathematics.</
               a>
%F A001158 Multiplicative with a(p^e) = (p^(3e+3)-1)/(p^3-1). - David W. Wilson 
               (davidwwilson(AT)comcast.net), Aug 01, 2001.
%F A001158 G.f.: sum(k>=1, k^3*x^k/(1-x^k)). - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Apr 21 2003
%F A001158 Equals A051731 * [1, 8, 27, 64, 125,...] = A127093 * [1, 4, 9, 16, 25,
               ...]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 02 2007
%t A001158 Table[DivisorSigma[3,n],{n,100}]
%o A001158 (PARI from Joerg Arndt (arndt(AT)jjj.de), May 03, 2008)
%o A001158 N=17; default(seriesprecision,N); x=z+O(z^(N+1))
%o A001158 c=sum(j=1,N,j*x^j); \\ log case
%o A001158 s=-log(prod(j=1,N,(1-x^j)^(j^2))); \\ A001158 sum of cubes of divisors 
               of n.
%o A001158 s=serconvol(s,c)
%o A001158 v=Vec(s)
%o A001158 (Other) sage: [sigma(n,3)for n in xrange(1,40)] # [From Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), Jun 04 2009]
%Y A001158 Cf. A000005, A000203, A001157.
%Y A001158 Cf. A051731, A127093.
%Y A001158 Sequence in context: A062451 A065959 A017669 this_sequence A053819 A085292 
               A073706
%Y A001158 Adjacent sequences: A001155 A001156 A001157 this_sequence A001159 A001160 
               A001161
%K A001158 nonn,easy,nice,mult
%O A001158 1,2
%A A001158 N. J. A. Sloane (njas(AT)research.att.com), R. K. Guy
%E A001158 Corrected Mathematica code T. D. Noe (noe(AT)sspectra.com), Mar 22 2009

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