A017687 - OEIS

%I #16 Apr 02 2024 02:56:57

%S 1,4097,531442,16781313,244140626,1088658937,13841287202,68736258049,

%T 282430067923,500122072361,3138428376722,1486382423891,23298085122482,

%U 28353876833297,129746582562692,281543712968705,582622237229762,1157115988280531,2213314919066162,2048500130460969

%N Numerator of sum of -12th 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="/A017687/b017687.txt">Table of n, a(n) for n = 1..10000</a>

%F From _Amiram Eldar_, Apr 02 2024: (Start)

%F sup_{n>=1} a(n)/A017688(n) = zeta(12) (A013670).

%F Dirichlet g.f. of a(n)/A017688(n): zeta(s)*zeta(s+12).

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017688(k) = zeta(13) (A013671). (End)

%t Table[Numerator[DivisorSigma[12, n]/n^12], {n, 1, 20}] (* _G. C. Greubel_, Nov 06 2018 *)

%o (PARI) vector(20, n, numerator(sigma(n, 12)/n^12)) \\ _G. C. Greubel_, Nov 06 2018

%o (Magma) [Numerator(DivisorSigma(12,n)/n^12): n in [1..20]]; // _G. C. Greubel_, Nov 06 2018

%Y Cf. A017688 (denominator), A013670, A013671.

%K nonn,frac

%O 1,2

%A _N. J. A. Sloane_