A017695 - OEIS

%I #20 Apr 02 2024 03:40:05

%S 1,65537,43046722,4295032833,152587890626,1410576509857,

%T 33232930569602,281479271743489,1853020231898563,5000076293978081,

%U 45949729863572162,30814514057170571,665416609183179842,1088993285370003137,6568408508343827972,18447025552981295105,48661191875666868482

%N Numerator of sum of -16th 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="/A017695/b017695.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)/A017696(n) = zeta(16) (A013674).

%F Dirichlet g.f. of a(n)/A017696(n): zeta(s)*zeta(s+16).

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017696(k) = zeta(17) (A013675). (End)

%t Table[Numerator[Total[1/Divisors[n]^16]],{n,20}] (* _Harvey P. Dale_, Sep 26 2014 *)

%t Table[Numerator[DivisorSigma[16, n]/n^16], {n, 1, 20}] (* _G. C. Greubel_, Nov 05 2018 *)

%o (PARI) vector(20, n, numerator(sigma(n, 16)/n^16)) \\ _G. C. Greubel_, Nov 05 2018

%o (Magma) [Numerator(DivisorSigma(16,n)/n^16): n in [1..20]]; // _G. C. Greubel_, Nov 05 2018

%Y Cf. A017696 (denominator), A013674, A013675.

%K nonn,frac

%O 1,2

%A _N. J. A. Sloane_