OFFSET
1,3
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
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, d<n} A001227(n/d) * a(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
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 *)
PROG
(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
(PARI) a(n)={sumdiv(n, d, if(d%2, moebius(d)*moebius(n/d)))} \\ Andrew Howroyd, Sep 23 2019
CROSSREFS
KEYWORD
sign,mult,easy
AUTHOR
Ilya Gutkovskiy, Sep 15 2019
STATUS
approved