A017701 - OEIS

%I #16 Apr 02 2024 08:52:38

%S 1,524289,1162261468,274878431233,19073486328126,50780075233021,

%T 11398895185373144,144115462954287105,1350851718835253557,

%U 5000009536743426207,61159090448414546292,79870152251600907511,1461920290375446110678,747039419730512536827,7389459406535218149656

%N Numerator of sum of -19th 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="/A017701/b017701.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)/A017702(n) = zeta(19) (A013677).

%F Dirichlet g.f. of a(n)/A017702(n): zeta(s)*zeta(s+19).

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017702(k) = zeta(20) (A013678). (End)

%t Table[Numerator[DivisorSigma[19, n]/n^19], {n, 1, 20}] (* _G. C. Greubel_, Nov 05 2018 *)

%o (PARI) vector(20, n, numerator(sigma(n, 19)/n^19)) \\ _G. C. Greubel_, Nov 05 2018

%o (Magma) [Numerator(DivisorSigma(19,n)/n^19): n in [1..20]]; // _G. C. Greubel_, Nov 05 2018

%Y Cf. A017702 (denominator), A013677, A013678.

%K nonn,frac

%O 1,2

%A _N. J. A. Sloane_