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%I M5041 N2177 #69 Jan 27 2024 05:25:41
%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(Product_{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 Pérez Herrero_, Jan 19 2013
%F G..f.: Sum_{n >= 1} A(4,x^n)/(1 - x^n)^5, where A(4,x) = x + 11*x^2 + 11*x^3 + x^4 is the 4th Eulerian polynomial - see A008292. - _Peter Bala_, Jan 11 2021
%F a(n) = Sum_{1 <= i, j, k, l <= n} tau(gcd(i, j, k, l, n)) = Sum_{d divides n} tau(d) * J_4(n/d), where the divisor function tau(n) = A000005(n) and the Jordan totient function J_4(n) = A059377(n). - _Peter Bala_, Jan 22 2024
%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 range(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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