A347992 - OEIS

%I #48 Oct 14 2021 08:48:53

%S 1,0,0,1,0,-2,0,0,1,-2,0,-2,0,-2,-2,1,0,-2,0,-2,-2,-2,0,-4,1,-2,0,-2,

%T 0,-6,0,0,-2,-2,-2,-1,0,-2,-2,-4,0,-6,0,-2,-2,-2,0,-4,1,-2,-2,-2,0,-4,

%U -2,-4,-2,-2,0,-8,0,-2,-2,1,-2,-6,0,-2,-2,-6,0,-4,0,-2,-2,-2,-2,-6,0,-4,1,-2,0,-8,-2,-2,-2

%N a(n) = Sum_{d|n} (-1)^(tau(d) - 1).

%H Seiichi Manyama, <a href="/A347992/b347992.txt">Table of n, a(n) for n = 1..10000</a>

%F If p is prime, a(p) = 0.

%F If p is prime, a(p^even) = 1 and a(p^odd) = 0. - _Michel Marcus_, Oct 13 2021

%F If p <> q primes, a(p*q) = -2 (A006881). - _Bernard Schott_, Oct 13 2021

%F G.f.: Sum_{k>=1} (-1)^(tau(k) - 1) * x^k/(1 - x^k). - _Seiichi Manyama_, Oct 14 2021

%t a[n_] := DivisorSum[n, (-1)^(DivisorSigma[0, #] - 1) &]; Array[a, 100] (* _Amiram Eldar_, Oct 08 2021 *)

%o (PARI) a(n) = sumdiv(n, d, (-1)^(numdiv(d)-1));

%o (PARI) my(N=99, x='x+O('x^N)); Vec(sum(k=1, N, (-1)^(numdiv(k)-1)*x^k/(1-x^k)))

%Y Cf. A000005 (tau), A001248, A006881, A347405, A348223.

%K sign

%O 1,6

%A _Seiichi Manyama_, Oct 08 2021