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A017689 Numerator of sum of -13th powers of divisors of n.

%I A017689 #18 Apr 02 2024 02:57:07
%S A017689 1,8193,1594324,67117057,1220703126,1088524711,96889010408,
%T A017689 549822930945,2541867422653,5000610355659,34522712143932,
%U A017689 26751583696117,302875106592254,99226457784093,648732096885608,4504149450301441,9904578032905938,6941839931265343,42052983462257060
%N A017689 Numerator of sum of -13th powers of divisors of n.
%C A017689 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 A017689 G. C. Greubel, <a href="/A017689/b017689.txt">Table of n, a(n) for n = 1..10000</a>
%F A017689 From _Amiram Eldar_, Apr 02 2024: (Start)
%F A017689 sup_{n>=1} a(n)/A017690(n) = zeta(13) (A013671).
%F A017689 Dirichlet g.f. of a(n)/A017690(n): zeta(s)*zeta(s+13).
%F A017689 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017690(k) = zeta(14) (A013672). (End)
%p A017689 A017689 := proc(n)
%p A017689     numtheory[sigma][-13](n) ;
%p A017689     numer(%) ;
%p A017689 end proc: # _R. J. Mathar_, Sep 21 2017
%t A017689 Table[Numerator[DivisorSigma[13, n]/n^13], {n, 1, 20}] (* _G. C. Greubel_, Nov 06 2018 *)
%o A017689 (PARI) vector(20, n, numerator(sigma(n, 13)/n^13)) \\ _G. C. Greubel_, Nov 06 2018
%o A017689 (Magma) [Numerator(DivisorSigma(13,n)/n^13): n in [1..20]]; // _G. C. Greubel_, Nov 06 2018
%Y A017689 Cf. A017690 (denominator), A013671, A013672.
%K A017689 nonn,frac
%O A017689 1,2
%A A017689 _N. J. A. Sloane_