A361984 a(1) = 1, a(2) = 0; a(n) = Sum_{d|n, d < n} (-1)^(n/d) a(d).

1, 0, -1, 1, -1, 0, -1, 2, 0, 0, -1, -1, -1, 0, 1, 4, -1, 0, -1, -1, 1, 0, -1, -2, 0, 0, 0, -1, -1, 0, -1, 8, 1, 0, 1, 0, -1, 0, 1, -2, -1, 0, -1, -1, 0, 0, -1, -4, 0, 0, 1, -1, -1, 0, 1, -2, 1, 0, -1, 1, -1, 0, 0, 16, 1, 0, -1, -1, 1, 0, -1, 0, -1, 0, 0, -1, 1, 0, -1, -4, 0, 0, -1, 1, 1, 0, 1, -2, -1, 0, 1, -1, 1, 0, 1, -8, -1, 0, 0, 0

Offset: 1

Author

Seiichi Manyama, Apr 02 2023

Keywords

• sign
• mult

Links

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

Crossrefs

Partial sums give A309288.

Cf. A361985, A361986.

Cf. A092673.

Programs

Mathematica

f[p_, e_] := If[e == 1, -1, 0]; f[2, e_] := If[e == 1, 0, 2^(e-2)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 09 2023 *)

Formula

a(n) is multiplicative with a(2) = 0, a(2^e) = 2^(e-2) if e>1. a(p) = -1, a(p^e) = 0 if e>1, p>2.

G.f. A(x) satisfies -x * (1 - x) = Sum_{k>=1} (-1)^k * A(x^k).