A355837 Dirichlet inverse of A322327.

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

Offset: 1

Author

Antti Karttunen, Jul 19 2022, based on Werner Schulte's comment in A322327

Keywords

• sign
• mult

Links

• Antti Karttunen, Table of n, a(n) for n = 1..100000
• Index entries for sequences computed from exponents in factorization of n

Crossrefs

Cf. A322327.

Programs

Mathematica

f[p_, e_] := 2 * (-1)^((e + 1)/2) * Mod[e, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jul 19 2022 *)

PARI

A355837(n) = factorback(apply(e -> 2*(e%2)*((-1)^((1+e)/2)), factor(n)[, 2]));

Formula

Multiplicative with a(p^e) = 2 * (e mod 2) * (-1)^((e+1)/2) for prime p and e>0.

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA322327(n/d) * a(d).

Dirichlet g.f.: zeta(4*s)/(zeta(s)^2*zeta(2*s)). - Amiram Eldar, Dec 29 2022