A328641 - OEIS

%I #12 Dec 03 2022 05:43:30

%S 1,-2,-4,-1,-8,8,-12,0,0,16,-20,4,-24,24,32,1,-32,0,-36,8,48,40,-44,0,

%T 8,48,4,12,-56,-64,-60,2,80,64,96,0,-72,72,96,0,-80,-96,-84,20,0,88,

%U -92,-4,24,-16,128,24,-104,-8,160,0,144,112,-116,-32,-120,120,0,3,192

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

%C Dirichlet inverse of A029935.

%C Dirichlet convolution of A023900 with itself.

%C Inverse Moebius transform of A101035.

%H Amiram Eldar, <a href="/A328641/b328641.txt">Table of n, a(n) for n = 1..10000</a>

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

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

%F 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

%t 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}]

%t 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 *)

%o (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

%Y Cf. A023900, A029935, A101035.

%K sign,mult

%O 1,2

%A _Ilya Gutkovskiy_, Oct 23 2019