cp's OEIS Frontend

A017681 Numerator of sum of -9th powers of divisors of n.

%I A017681 #22 Apr 02 2024 02:56:20
%S A017681 1,513,19684,262657,1953126,93499,40353608,134480385,387440173,
%T A017681 500976819,2357947692,1292535097,10604499374,2587675113,1423901192,
%U A017681 68853957121,118587876498,7361363287,322687697780,256501107891,113474345696,302406791499,1801152661464,24510295355
%N A017681 Numerator of sum of -9th powers of divisors of n.
%C A017681 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 A017681 G. C. Greubel, <a href="/A017681/b017681.txt">Table of n, a(n) for n = 1..10000</a>
%F A017681 Numerators of coefficients in expansion of Sum_{k>=1} x^k/(k^9*(1 - x^k)). - _Ilya Gutkovskiy_, May 25 2018
%F A017681 From _Amiram Eldar_, Apr 02 2024: (Start)
%F A017681 sup_{n>=1} a(n)/A017682(n) = zeta(9) (A013667).
%F A017681 Dirichlet g.f. of a(n)/A017682(n): zeta(s)*zeta(s+9).
%F A017681 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017682(k) = zeta(10) (A013668). (End)
%e A017681 1, 513/512, 19684/19683, 262657/262144, 1953126/1953125, 93499/93312, 40353608/40353607, 134480385/134217728, ...
%t A017681 Table[Numerator[Total[1/Divisors[n]^9]],{n,20}] (* _Harvey P. Dale_, Aug 26 2013 *)
%t A017681 Table[Numerator[DivisorSigma[9, n]/n^9], {n, 1, 20}] (* _G. C. Greubel_, Nov 07 2018 *)
%o A017681 (PARI) vector(20, n, numerator(sigma(n, 9)/n^9)) \\ _G. C. Greubel_, Nov 07 2018
%o A017681 (Magma) [Numerator(DivisorSigma(9,n)/n^9): n in [1..20]]; // _G. C. Greubel_, Nov 07 2018
%Y A017681 Cf. A017682 (denominator), A013667, A013668.
%K A017681 nonn,frac
%O A017681 1,2
%A A017681 _N. J. A. Sloane_