A017705 - OEIS

%I #17 Apr 02 2024 08:52:02

%S 1,2097153,10460353204,4398048608257,476837158203126,609360030634117,

%T 558545864083284008,9223376434903384065,109418989141972712413,

%U 500000238418580150139,7400249944258160101212,11501285462682212701357,247064529073450392704414,146419516812481413403653

%N Numerator of sum of -21st 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="/A017705/b017705.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)/A017706(n) = zeta(21).

%F Dirichlet g.f. of a(n)/A017706(n): zeta(s)*zeta(s+21).

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017706(k) = zeta(22). (End)

%t Table[Numerator[DivisorSigma[21, n]/n^21], {n, 1, 20}] (* _G. C. Greubel_, Nov 05 2018 *)

%o (PARI) vector(20, n, numerator(sigma(n, 21)/n^21)) \\ _G. C. Greubel_, Nov 05 2018

%o (Magma) [Numerator(DivisorSigma(21,n)/n^21): n in [1..20]]; // _G. C. Greubel_, Nov 05 2018

%Y Cf. A017706 (denominator).

%K nonn,frac

%O 1,2

%A _N. J. A. Sloane_