A010555 - OEIS

A010555

a(n) = 1 if n is the product of an even number of distinct primes, otherwise a(n) = -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

1,1

COMMENTS

The old definition was mobius(mobius(n)). The reason for this sequence was that in some ancient version of Maple the command "with(numtheory): A010555 := n->mobius(mobius(n));" worked. But it no longer works, so the definition has been changed.

LINKS

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

FORMULA

a(n) = (-1)^(1-A353629(n)). - Antti Karttunen, Jan 17 2025

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 6/Pi^2 - 1 = -0.392072... . - Amiram Eldar, Sep 17 2026

MATHEMATICA

a[n_] := If[MoebiusMu[n] == 1, 1, -1]; Array[a, 100] (* Amiram Eldar, Sep 17 2026 *)

PROG

(PARI) A010555(n) = if(1==moebius(n), 1, -1); \\ Antti Karttunen, Jan 17 2025

CROSSREFS

Cf. A008683, A030229 (positions of positive terms), A353629 (their characteristic function).

Sequence in context: A119664 A257075 A330034 * A108784 A127252 A244513