A327276 - OEIS

%I #28 Sep 08 2022 08:46:24

%S 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,

%T -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,

%U -2,0,-2,2,-2,0,4,-4,-2,0,4,-4,-2,0,-2,2,-2,0,4,-4,-2

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

%C Dirichlet inverse of A001227.

%C All terms are 0 or +/- a power of 2. - _Robert Israel_, Nov 26 2019

%H Robert Israel, <a href="/A327276/b327276.txt">Table of n, a(n) for n = 1..10000</a>

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

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

%F a(1) = 1; a(n) = -Sum_{d|n, d<n} A001227(n/d) * a(d).

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

%F 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

%p f:= proc(n) local m, d;

%p   m:= n/2^padic:-ordp(n,2);

%p   add(numtheory:-mobius(d)*numtheory:-mobius(n/d), d = numtheory:-divisors(m))

%p end proc:

%p map(f, [$1..100]); # _Robert Israel_, Nov 26 2019

%t Table[DivisorSum[n, MoebiusMu[#] MoebiusMu[n/#] &, OddQ[#] &], {n, 1, 79}]

%t 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}]

%t 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 *)

%o (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

%o (PARI) a(n)={sumdiv(n, d, if(d%2, moebius(d)*moebius(n/d)))} \\ _Andrew Howroyd_, Sep 23 2019

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

%K sign,mult,easy

%O 1,3

%A _Ilya Gutkovskiy_, Sep 15 2019