A355691 Dirichlet inverse of A320111, number of divisors of n that are not of the form 4k+2.

1, -1, -2, -1, -2, 2, -2, 0, 1, 2, -2, 2, -2, 2, 4, 1, -2, -1, -2, 2, 4, 2, -2, 0, 1, 2, 0, 2, -2, -4, -2, 1, 4, 2, 4, -1, -2, 2, 4, 0, -2, -4, -2, 2, -2, 2, -2, -2, 1, -1, 4, 2, -2, 0, 4, 0, 4, 2, -2, -4, -2, 2, -2, 0, 4, -4, -2, 2, 4, -4, -2, 0, -2, 2, -2, 2, 4, -4, -2, -2, 0, 2, -2, -4, 4, 2, 4, 0, -2, 2, 4, 2, 4, 2, 4, -2, -2, -1, -2, -1, -2, -4, -2, 0, -8

Offset: 1

Author

Antti Karttunen, Jul 15 2022

Keywords

• sign
• mult

Comments

Multiplicative because A320111 is.

Links

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

Crossrefs

Cf. A320111.

Cf. also A355690.

Cf. A010892.

Programs

Mathematica

f[2, e_] := Switch[Mod[e, 6], 0, 0, 1, -1, 2, -1, 3, 0, 4, 1, 5, 1]; f[p_, 1] = -2; f[p_, 2] = 1; f[p_, e_] := 0; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 30 2022 *)

PARI

A320111(n) = sumdiv(n,d,(2!=(d%4)));
memoA355691 = Map();
A355691(n) = if(1==n,1,my(v); if(mapisdefined(memoA355691,n,&v), v, v = -sumdiv(n,d,if(dA320111(n/d)*A355691(d),0)); mapput(memoA355691,n,v); (v)));

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA320111(n/d) * a(d).

Multiplicative with a(2^e) = A010892(e+2) and for a prime p > 2, a(p) = -2, a(p^2) = 1 and a(p^e) = 0 when e > 2. - Sebastian Karlsson, Oct 21 2022

Dirichlet g.f.: 4^s/(zeta(s)^2*(1 - 2^s + 4^s)). - Amiram Eldar, Dec 30 2022