A349340 Dirichlet inverse of A003557, where A003557 is multiplicative with a(p^e) = p^(e-1).

1, -1, -1, -1, -1, 1, -1, -1, -2, 1, -1, 1, -1, 1, 1, -1, -1, 2, -1, 1, 1, 1, -1, 1, -4, 1, -4, 1, -1, -1, -1, -1, 1, 1, 1, 2, -1, 1, 1, 1, -1, -1, -1, 1, 2, 1, -1, 1, -6, 4, 1, 1, -1, 4, 1, 1, 1, 1, -1, -1, -1, 1, 2, -1, 1, -1, -1, 1, 1, -1, -1, 2, -1, 1, 4, 1, 1, -1, -1, 1, -8, 1, -1, -1, 1, 1, 1, 1, -1, -2, 1

Offset: 1

Author

Antti Karttunen, Nov 18 2021

Keywords

• sign
• mult

Links

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

Crossrefs

Cf. A003557, A076479, A326297 (absolute values).

Cf. also A325126, A349350, A349619.

Programs

Mathematica

f[p_, e_] := -(p - 1)^(e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 18 2021 *)

PARI

A003557(n) = (n/factorback(factorint(n)[, 1]));
memoA349340 = Map();
A349340(n) = if(1==n,1,my(v); if(mapisdefined(memoA349340,n,&v), v, v = -sumdiv(n,d,if(dA003557(n/d)*A349340(d),0)); mapput(memoA349340,n,v); (v)));

PARI

A349340(n) = { my(f=factor(n)); prod(i=1, #f~, -((f[i,1]-1)^(f[i,2]-1))); };

Formula

Multiplicative with a(p^e) = -((p-1)^(e-1)).

a(n) = A076479(n) * A326297(n).

a(1) = 1; a(n) = -Sum_{d|n, d < n} A003557(n/d) * a(d).