A328641 Dirichlet g.f.: zeta(s)^2 / zeta(s-1)^2.

1, -2, -4, -1, -8, 8, -12, 0, 0, 16, -20, 4, -24, 24, 32, 1, -32, 0, -36, 8, 48, 40, -44, 0, 8, 48, 4, 12, -56, -64, -60, 2, 80, 64, 96, 0, -72, 72, 96, 0, -80, -96, -84, 20, 0, 88, -92, -4, 24, -16, 128, 24, -104, -8, 160, 0, 144, 112, -116, -32, -120, 120, 0, 3, 192

Offset

1,2

Comments

Dirichlet inverse of A029935.

Dirichlet convolution of A023900 with itself.

Inverse Moebius transform of A101035.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

a(1) = 1; a(n) = -Sum_{d|n, d<n} A029935(n/d) * a(d).

a(n) = Sum_{d|n} A101035(d).

Multiplicative with a(p) = 2*(1-p), and a(p^e) = (p-1)*(e*p-p-e-1) for e > 1. - Amiram Eldar, Dec 03 2022

Mathematica

a[1] = 1; a[n_] := -Sum[DirichletConvolve[EulerPhi[j], EulerPhi[j], j, n/d] a[d], {d, Most @ Divisors[n]}]; Table[a[n], {n, 1, 65}]
f[p_, e_] := If[e == 1, 2*(1 - p), (p - 1)*(e*p - p - e - 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 03 2022 *)

Prog

(PARI) seq(n)={dirdiv(vector(n, n, n==1), vector(n, n, sumdiv(n, d, eulerphi(d) * eulerphi(n/d))))} \\ Andrew Howroyd, Oct 25 2019

Crossrefs

Cf. A023900, A029935, A101035.

Sequence in context: A127151 A193034 A254179 * A208741 A335957 A057115

Adjacent sequences: A328638 A328639 A328640 * A328642 A328643 A328644

Keyword

sign,mult

Author

Ilya Gutkovskiy, Oct 23 2019

Status

approved