A370297 Inverse Moebius transform of A322328.

1, 5, 5, 13, 5, 25, 5, 25, 13, 25, 5, 65, 5, 25, 25, 41, 5, 65, 5, 65, 25, 25, 5, 125, 13, 25, 25, 65, 5, 125, 5, 61, 25, 25, 25, 169, 5, 25, 25, 125, 5, 125, 5, 65, 65, 25, 5, 205, 13, 65, 25, 65, 5, 125, 25, 125, 25, 25, 5, 325, 5, 25, 65, 85, 25, 125, 5, 65, 25, 125, 5, 325

Offset: 1

Author

Werner Schulte, Feb 14 2024

Keywords

• nonn
• easy
• mult

Crossrefs

Cf. A034444, A048691, A322328.

Programs

Mathematica

f[p_, e_] := 2*e^2 + 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+2*e*(e+1), factor(n)[,2]))

Formula

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

Dirichlet g.f.: (zeta(s))^5 / (zeta(2*s))^2.

Dirichlet convolution of A034444 and A048691.

Dirichlet inverse sequence b(n) for n > 0 is multiplicative with b(p) = -5 and b(p^e) = (-1)^e * (8*e-4) for prime p and e > 1.