%I A001159 M5041 N2177
%S A001159 1,17,82,273,626,1394,2402,4369,6643,10642,14642,22386,28562,40834,
%T A001159 51332,69905,83522,112931,130322,170898,196964,248914,279842,358258,
%U A001159 391251,485554,538084,655746,707282,872644,923522,1118481,1200644
%N A001159 sigma_4(n): sum of 4th powers of divisors of n.
%C A001159 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 A001159 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 A001159 sigma_4(n) is the sum of the 4th powers of the divisors of n (A001159).
%D A001159 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 A001159 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 
               1976, page 38.
%D A001159 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001159 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A001159 T. D. Noe, <a href="b001159.txt">Table of n, a(n) for n = 1..10000</a>
%H A001159 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 A001159 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.
%F A001159 Multiplicative with a(p^e) = (p^(4e+4)-1)/(p^4-1). - David W. Wilson 
               (davidwwilson(AT)comcast.net), Aug 01, 2001.
%F A001159 G.f. sum(k>=1, k^4*x^k/(1-x^k)). - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Apr 21 2003
%t A001159 lst={};Do[AppendTo[lst,DivisorSigma[4,n]],{n,5!}];lst [From Vladimir 
               Orlovsky (4vladimir(AT)gmail.com), Mar 11 2009]
%o A001159 (PARI from Joerg Arndt (arndt(AT)jjj.de), May 03, 2008)
%o A001159 N=17; default(seriesprecision,N); x=z+O(z^(N+1))
%o A001159 c=sum(j=1,N,j*x^j); \\ log case
%o A001159 s=-log(prod(j=1,N,(1-x^j)^(j^3))); \\ A001159 sigma_4(n)
%o A001159 s=serconvol(s,c)
%o A001159 v=Vec(s)
%o A001159 (Other) sage: [sigma(n,4)for n in xrange(1,34)] # [From Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), Jun 04 2009]
%Y A001159 Cf. A000005, A000203, A001157, A001158.
%Y A001159 Sequence in context: A034678 A065960 A017671 this_sequence A053820 A142059 
               A158528
%Y A001159 Adjacent sequences: A001156 A001157 A001158 this_sequence A001160 A001161 
               A001162
%K A001159 nonn,easy,mult
%O A001159 1,2
%A A001159 N. J. A. Sloane (njas(AT)research.att.com).