A327276 a(n) = Sum_{d|n, d odd} mu(d) * mu(n/d).

1, -1, -2, 0, -2, 2, -2, 0, 1, 2, -2, 0, -2, 2, 4, 0, -2, -1, -2, 0, 4, 2, -2, 0, 1, 2, 0, 0, -2, -4, -2, 0, 4, 2, 4, 0, -2, 2, 4, 0, -2, -4, -2, 0, -2, 2, -2, 0, 1, -1, 4, 0, -2, 0, 4, 0, 4, 2, -2, 0, -2, 2, -2, 0, 4, -4, -2, 0, 4, -4, -2, 0, -2, 2, -2, 0, 4, -4, -2

Offset: 1

Author

Ilya Gutkovskiy, Sep 15 2019

Keywords

• sign
• mult
• easy

Comments

Dirichlet inverse of A001227.

All terms are 0 or +/- a power of 2. - Robert Israel, Nov 26 2019

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001227, A007427, A008683, A068068, A209229, A327278.

Programs

Magma

[&+[MoebiusMu(d)*MoebiusMu(n div d): d in [a:a in Divisors(n)| IsOdd(a)]]:n in [1..80]]; // Marius A. Burtea, Sep 15 2019

Maple

f:= proc(n) local m, d;
  m:= n/2^padic:-ordp(n,2);
  add(numtheory:-mobius(d)*numtheory:-mobius(n/d), d = numtheory:-divisors(m))
end proc:
map(f, [$1..100]); # Robert Israel, Nov 26 2019

Mathematica

Table[DivisorSum[n, MoebiusMu[#] MoebiusMu[n/#] &, OddQ[#] &], {n, 1, 79}]
a[n_] := If[n == 1, n, -Sum[If[d < n, DivisorSum[n/d, Mod[#, 2] &] a[d], 0], {d, Divisors[n]}]]; Table[a[n], {n, 1, 79}]
f[p_, e_] := Which[e == 1, -1 - Boole[p > 2], e == 2, Boole[p > 2], e > 2, 0]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 25 2020 *)

PARI

a(n)={sumdiv(n, d, if(d%2, moebius(d)*moebius(n/d)))} \\ Andrew Howroyd, Sep 23 2019

Formula

G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} A001227(k) * A(x^k).

Dirichlet g.f.: 1 / (zeta(s)^2 * (1 - 1/2^s)).

a(1) = 1; a(n) = -Sum_{d|n, dA001227(n/d) * a(d).

a(n) = Sum_{d|n} A209229(n/d) * A007427(d).

Multiplicative with a(2^e) = -1 if e = 1, and 0 if e > 1, and a(p^e) = -2 if e = 1, 1 if e = 2, and 0 if e > 2, for an odd prime p. - Amiram Eldar, Oct 25 2020