A017691 - OEIS

%I #16 Apr 02 2024 04:09:48

%S 1,16385,4782970,268451841,6103515626,39184481725,678223072850,

%T 4398314962945,22876797237931,10000610353201,379749833583242,

%U 213999516991295,3937376385699290,5556342524323625,5838586426737844,72061992352890881,168377826559400930,374836322743499435

%N Numerator of sum of -14th 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="/A017691/b017691.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)/A017692(n) = zeta(14) (A013672).

%F Dirichlet g.f. of a(n)/A017692(n): zeta(s)*zeta(s+14).

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017692(k) = zeta(15) (A013673). (End)

%t Table[Numerator[DivisorSigma[14, n]/n^14], {n, 1, 20}] (* _G. C. Greubel_, Nov 06 2018 *)

%o (PARI) vector(20, n, numerator(sigma(n, 14)/n^14)) \\ _G. C. Greubel_, Nov 06 2018

%o (Magma) [Numerator(DivisorSigma(14,n)/n^14): n in [1..20]]; // _G. C. Greubel_, Nov 06 2018

%Y Cf. A017692 (denominator), A013672, A013673.

%K nonn,frac

%O 1,2

%A _N. J. A. Sloane_