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%I #22 Apr 02 2024 02:55:56
%S 1,129,2188,16513,78126,23521,823544,2113665,4785157,5039127,19487172,
%T 9032611,62748518,13279647,56979896,270549121,410338674,205761751,
%U 893871740,645047319,1801914272,628461297,3404825448,385391585,6103593751,4047279411,10465138360,34691791
%N Numerator of sum of -7th 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="/A017677/b017677.txt">Table of n, a(n) for n = 1..10000</a>
%F Numerators of coefficients in expansion of Sum_{k>=1} x^k/(k^7*(1 - x^k)). - _Ilya Gutkovskiy_, May 25 2018
%F From _Amiram Eldar_, Apr 02 2024: (Start)
%F sup_{n>=1} a(n)/A017678(n) = zeta(7) (A013665).
%F Dirichlet g.f. of a(n)/A017678(n): zeta(s)*zeta(s+7).
%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017678(k) = zeta(8) (A013666). (End)
%e 1, 129/128, 2188/2187, 16513/16384, 78126/78125, 23521/23328, 823544/823543, 2113665/2097152, ...
%t Table[Numerator[Total[Divisors[n]^-7]],{n,30}] (* _Harvey P. Dale_, Nov 29 2014 *)
%t Table[Numerator[DivisorSigma[7, n]/n^7], {n, 1, 20}] (* _G. C. Greubel_, Nov 07 2018 *)
%o (PARI) vector(20, n, numerator(sigma(n, 7)/n^7)) \\ _G. C. Greubel_, Nov 07 2018
%o (Magma) [Numerator(DivisorSigma(7,n)/n^7): n in [1..20]]; // _G. C. Greubel_, Nov 07 2018
%Y Cf. A017678 (denominator), A013665, A013666.
%K nonn,frac
%O 1,2
%A _N. J. A. Sloane_
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