Voozh

A359154

a(n) = (-1)^sopfr(n), where sopfr is the sum of prime factors with repetition.

1, 1, -1, 1, -1, -1, -1, 1, 1, -1, -1, -1, -1, -1, 1, 1, -1, 1, -1, -1, 1, -1, -1, -1, 1, -1, -1, -1, -1, 1, -1, 1, 1, -1, 1, 1, -1, -1, 1, -1, -1, 1, -1, -1, -1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, -1, 1, -1, -1, 1, -1, -1, -1, 1, 1, 1, -1, -1, 1, 1, -1, 1, -1, -1, -1, -1, 1, 1, -1, -1, 1, -1, -1, 1, 1, -1, 1, -1, -1, -1, 1, -1, 1, -1, 1, -1, -1, 1, -1, 1, -1, 1, -1, -1, -1

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OFFSET

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..100000

FORMULA

Multiplicative with a(p^e) = (-1)^(p*e). Fully multiplicative with a(p) = (-1)^p.

a(n) = (-1)^A001414(n) = (-1)^(1+A356163(n)).

For all n >= 1, a(A003961(n)) = A008836(n).

Dirichlet g.f.: (zeta(2*s)/zeta(s))*((2^s+1)/(2^s-1)). - Amiram Eldar, Dec 29 2022

MATHEMATICA

f[p_, e_] := (-1)^(p*e); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 29 2022 *)

PROG