A370296 Inverse Moebius transform of A322327.

1, 3, 3, 7, 3, 9, 3, 13, 7, 9, 3, 21, 3, 9, 9, 21, 3, 21, 3, 21, 9, 9, 3, 39, 7, 9, 13, 21, 3, 27, 3, 31, 9, 9, 9, 49, 3, 9, 9, 39, 3, 27, 3, 21, 21, 9, 3, 63, 7, 21, 9, 21, 3, 39, 9, 39, 9, 9, 3, 63, 3, 9, 21, 43, 9, 27, 3, 21, 9, 27, 3, 91, 3, 9, 21, 21, 9, 27, 3, 63, 21, 9, 3, 63

Offset: 1

Author

Werner Schulte, Feb 14 2024

Keywords

• nonn
• easy
• mult

Crossrefs

Cf. A322327, A000005, A323308.

Programs

Mathematica

f[p_, e_] := e^2 + e + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 14 2024 *)

PARI

a(n) = factorback(apply(e->1+e+e^2,factor(n)[,2]))

Formula

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

Dirichlet g.f.: (zeta(s))^3 * zeta(2*s) / zeta(4*s).

Dirichlet inverse sequence b(n) for n > 0 is multiplicative with b(p) = -3 and b(p^e) = 2 * (-1)^((e+1)*(e+2)/2) for prime p and e > 1.

Dirichlet convolution of A000005 and A323308.