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A017671 Numerator of sum of -4th powers of divisors of n.

%I A017671 #23 Apr 02 2024 02:55:30
%S A017671 1,17,82,273,626,697,2402,4369,6643,5321,14642,3731,28562,20417,51332,
%T A017671 69905,83522,112931,130322,85449,196964,124457,279842,179129,391251,
%U A017671 242777,538084,46839,707282,218161,923522,1118481,1200644,41761,1503652,604513,1874162
%N A017671 Numerator of sum of -4th powers of divisors of n.
%C A017671 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
%H A017671 G. C. Greubel, <a href="/A017671/b017671.txt">Table of n, a(n) for n = 1..10000</a>
%F A017671 Numerators of coefficients in expansion of Sum_{k>=1} x^k/(k^4*(1 - x^k)). - _Ilya Gutkovskiy_, May 24 2018
%F A017671 From _Amiram Eldar_, Apr 02 2024: (Start)
%F A017671 sup_{n>=1} a(n)/A017672(n) = zeta(4) (A013662).
%F A017671 Dirichlet g.f. of a(n)/A017672(n): zeta(s)*zeta(s+4).
%F A017671 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017672(k) = zeta(5) (A013663). (End)
%e A017671 1, 17/16, 82/81, 273/256, 626/625, 697/648, 2402/2401, 4369/4096, 6643/6561, 5321/5000, ...
%t A017671 Table[Numerator[DivisorSigma[-4, n]], {n, 50}] (* _Vladimir Joseph Stephan Orlovsky_, Jul 21 2011 *)
%t A017671 Table[Numerator[DivisorSigma[4, n]/n^4], {n, 1, 40}] (* _G. C. Greubel_, Nov 08 2018 *)
%o A017671 (PARI) vector(40, n, numerator(sigma(n, 4)/n^4)) \\ _G. C. Greubel_, Nov 08 2018
%o A017671 (Magma) [Numerator(DivisorSigma(4,n)/n^4): n in [1..40]]; // _G. C. Greubel_, Nov 08 2018
%Y A017671 Cf. A017672 (denominator), A013662, A013663.
%K A017671 nonn,frac
%O A017671 1,2
%A A017671 _N. J. A. Sloane_