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A001159 sigma_4(n): sum of 4th powers of divisors of n.
(Formerly M5041 N2177)
50

%I M5041 N2177

%S 1,17,82,273,626,1394,2402,4369,6643,10642,14642,22386,28562,40834,

%T 51332,69905,83522,112931,130322,170898,196964,248914,279842,358258,

%U 391251,485554,538084,655746,707282,872644,923522,1118481,1200644

%N sigma_4(n): sum of 4th powers of divisors of n.

%C 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 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. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001

%C sigma_4(n) is the sum of the 4th powers of the divisors of n (A001159).

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

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A001159/b001159.txt">Table of n, a(n) for n = 1..10000</a>

%H 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 [alternative scanned copy].

%H 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 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>

%F Multiplicative with a(p^e) = (p^(4e+4)-1)/(p^4-1). - _David W. Wilson_, Aug 01, 2001.

%F G.f. sum(k>=1, k^4*x^k/(1-x^k)). - _Benoit Cloitre_, Apr 21 2003

%F L.g.f.: -log(prod(j>=1 ,(1-x^j)^(j^3))) = Sum_{n>=1} a(n)/n*x^n. - _Joerg Arndt_, Feb 04 2011

%F Dirichlet g.f.: zeta(s)*zeta(s-4). - _R. J. Mathar_, Feb 04 2011

%F a(n) = Sum_{d|n} tau_{-2}^(d)*J_4(n/d), where tau_{-2} is A007427 and J_4 A059377. - _Enrique PĂ©rez Herrero_, Jan 19 2013

%p with(numtheory); A001159 := proc(n) sigma[4](n) ; end proc: # _R. J. Mathar_, Feb 04 2011

%t lst={}; Do[AppendTo[lst, DivisorSigma[4,n]], {n,5!}]; lst (* _Vladimir Joseph Stephan Orlovsky_, Mar 11 2009 *)

%t DivisorSigma[4,Range[40]] (* _Harvey P. Dale_, Apr 28 2013 *)

%o (PARI) N=99;q='q+O('q^N);

%o Vec(sum(n=1,N,n^4*q^n/(1-q^n))) /* _Joerg Arndt_, Feb 04 2011 */

%o (Sage) [sigma(n,4)for n in xrange(1,34)] # Zerinvary Lajos_, Jun 04 2009

%o (Maxima) makelist(divsum(n,4),n,1,100); /* _Emanuele Munarini_, Mar 26 2011 */

%o (MAGMA) [DivisorSigma(4,n): n in [1..40]]; // _Bruno Berselli_, Apr 10 2013

%Y Cf. A000005, A000203, A001157, A001158.

%K nonn,easy,mult

%O 1,2

%A _N. J. A. Sloane_

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Last modified November 14 15:22 EST 2019. Contains 329126 sequences. (Running on oeis4.)