A347992 a(n) = Sum_{d|n} (-1)^(tau(d) - 1).

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, 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, -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

Offset: 1

Author

Seiichi Manyama, Oct 08 2021

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000

Crossrefs

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

Programs

Mathematica

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

PARI

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

PARI

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

Formula

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

If p is prime, a(p^even) = 1 and a(p^odd) = 0. - Michel Marcus, Oct 13 2021

If p <> q primes, a(p*q) = -2 (A006881). - Bernard Schott, Oct 13 2021

G.f.: Sum_{k>=1} (-1)^(tau(k) - 1) * x^k/(1 - x^k). - Seiichi Manyama, Oct 14 2021