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%I #30 Apr 02 2024 02:55:18
%S 1,9,28,73,126,7,344,585,757,567,1332,511,2198,387,392,4681,4914,757,
%T 6860,4599,1376,2997,12168,455,15751,9891,20440,3139,24390,147,29792,
%U 37449,4144,22113,6192,55261,50654,15435,61544,7371,68922,172,79508,24309,10598
%N Numerator of sum of -3rd 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="/A017669/b017669.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Vincenzo Librandi)
%F Numerators of coefficients in expansion of Sum_{k>=1} x^k/(k^3*(1 - x^k)). - _Ilya Gutkovskiy_, May 24 2018
%F From _Amiram Eldar_, Apr 02 2024: (Start)
%F sup_{n>=1} a(n)/A017670(n) = zeta(3) (A002117).
%F Dirichlet g.f. of a(n)/A017670(n): zeta(s)*zeta(s+3).
%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017670(k) = zeta(4) (A013662). (End)
%e 1, 9/8, 28/27, 73/64, 126/125, 7/6, 344/343, 585/512, 757/729, 567/500, 1332/1331, 511/432, ...
%t Table[Numerator[DivisorSigma[-3, n]], {n, 50}] (* _Vladimir Joseph Stephan Orlovsky_, Jul 21 2011 *)
%t Table[Numerator[DivisorSigma[3, n]/n^3], {n, 1, 40}] (* _G. C. Greubel_, Nov 08 2018 *)
%o (PARI) vector(40, n, numerator(sigma(n, 3)/n^3)) \\ _G. C. Greubel_, Nov 08 2018
%o (Magma) [Numerator(DivisorSigma(3,n)/n^3): n in [1..40]]; // _G. C. Greubel_, Nov 08 2018
%Y Cf. A017670 (denominator), A002117, A013662.
%K nonn,frac
%O 1,2
%A _N. J. A. Sloane_
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