A297381 - OEIS

%I #20 Sep 30 2018 20:25:19

%S -1,1,1,1,2,-1,3,1,1,-2,5,-1,6,-3,-4,1,8,-1,9,-2,-6,-5,11,-1,2,-6,1,

%T -3,14,4,15,1,-10,-8,-12,-1,18,-9,-12,-2,20,6,21,-5,-4,-11,23,-1,3,-2,

%U -16,-6,26,-1,-20,-3,-18,-14,29,4,30,-15,-6,1,-24,10,33,-8,-22,12,35,-1,36,-18,-4,-9,-30,12,39,-2,1,-20,41,6,-32,-21,-28,-5

%N Numerator of -A023900(n)/2.

%H Antti Karttunen, <a href="/A297381/b297381.txt">Table of n, a(n) for n = 1..65537</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Ces%C3%A0ro_summation">Cesàro summation</a>

%F a(n) = numerator of -A023900(n)/2.

%F a(n) = numerator of lim_{s->0} zeta(s)*Sum_{d|n} A008683(d)/d^(s-1).

%F a(n) = numerator of lim_{N->infinity} (1/N)*Sum_{m=1..N} Sum_{k=1..m} A191898(n, k) for n > 1.

%F a(k) = numerators of lim_{N->infinity} (1/N)*Sum_{m=1..N} Sum_{n=1..m} A191898(n, k) for k > 1.

%t Clear[n, s, nn]; nn = 64; Numerator[Table[Limit[Zeta[s]*Total[MoebiusMu[Divisors[n]]/Divisors[n]^(s - 1)], s -> 0], {n, 1, nn}]]

%o (PARI) a(n) = numerator(-sumdiv(n, d, d*moebius(d))/2) \\ _Iain Fox_, Dec 29 2017

%o (PARI) A297381(n) = numerator(-(1/2)*factorback(apply(p -> 1-p, factor(n)[, 1]))); \\ _Antti Karttunen_, Sep 30 2018

%Y Cf. A023900, A297382 (denominators).

%K sign,frac

%O 1,5

%A _Mats Granvik_, Dec 29 2017

%E More terms from _Antti Karttunen_, Sep 30 2018