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%I #20 Apr 02 2024 02:56:08
%S 1,257,6562,65793,390626,843217,5764802,16843009,43053283,50195441,
%T 214358882,71955611,815730722,740777057,2563287812,4311810305,
%U 6975757442,11064693731,16983563042,12850228209,37828630724,27545116337,78310985282,55261912529,152588281251
%N Numerator of sum of -8th powers of divisors of n.
%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
%H G. C. Greubel, <a href="/A017679/b017679.txt">Table of n, a(n) for n = 1..10000</a>
%F Numerators of coefficients in expansion of Sum_{k>=1} x^k/(k^8*(1 - x^k)). - _Ilya Gutkovskiy_, May 25 2018
%F From _Amiram Eldar_, Apr 02 2024: (Start)
%F sup_{n>=1} a(n)/A017680(n) = zeta(8) (A013666).
%F Dirichlet g.f. of a(n)/A017680(n): zeta(s)*zeta(s+8).
%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017680(k) = zeta(9) (A013667). (End)
%e 1, 257/256, 6562/6561, 65793/65536, 390626/390625, 843217/839808, 5764802/5764801, 16843009/16777216, ...
%t Table[Numerator[DivisorSigma[8, n]/n^8], {n, 1, 20}] (* _G. C. Greubel_, Nov 07 2018 *)
%o (PARI) vector(20, n, numerator(sigma(n, 8)/n^8)) \\ _G. C. Greubel_, Nov 07 2018
%o (Magma) [Numerator(DivisorSigma(8,n)/n^8): n in [1..20]]; // _G. C. Greubel_, Nov 07 2018
%Y Cf. A017680 (denominator), A013666, A013667.
%K nonn,frac
%O 1,2
%A _N. J. A. Sloane_
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