A361019 Dirichlet inverse of A038712.

1, -3, -1, 2, -1, 3, -1, 0, 0, 3, -1, -2, -1, 3, 1, 0, -1, 0, -1, -2, 1, 3, -1, 0, 0, 3, 0, -2, -1, -3, -1, 0, 1, 3, 1, 0, -1, 3, 1, 0, -1, -3, -1, -2, 0, 3, -1, 0, 0, 0, 1, -2, -1, 0, 1, 0, 1, 3, -1, 2, -1, 3, 0, 0, 1, -3, -1, -2, 1, -3, -1, 0, -1, 3, 0, -2, 1, -3, -1, 0, 0, 3, -1, 2, 1, 3, 1, 0, -1, 0, 1, -2, 1, 3, 1, 0, -1

Offset: 1

Author

Antti Karttunen, Mar 02 2023

Keywords

• sign
• mult

Comments

Multiplicative because A038712 is.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A038712.

Programs

Mathematica

f[p_, e_] := If[e == 1, -1, 0]; f[2, e_] := If[e < 3, If[e == 1, -3, 2], 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Mar 02 2023 *)

PARI

A038712(n) = ((1<<(1+valuation(n,2)))-1);
memoA361019 = Map();
A361019(n) = if(1==n,1,my(v); if(mapisdefined(memoA361019,n,&v), v, v = -sumdiv(n,d,if(dA038712(n/d)*A361019(d),0)); mapput(memoA361019,n,v); (v)));

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA038712(n/d) * a(d).

Multiplicative with a(2) = -3, a(2^2) = 2, and a(2^e) = 0 for e > 2, and for odd prime p, a(p)= -1 and a(p^e) = 0 for e > 1. - Amiram Eldar, Mar 02 2023