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A323363 Dirichlet inverse of Dedekind's psi, A001615.

%I A323363 #22 Dec 05 2022 04:45:41
%S A323363 1,-3,-4,3,-6,12,-8,-3,4,18,-12,-12,-14,24,24,3,-18,-12,-20,-18,32,36,
%T A323363 -24,12,6,42,-4,-24,-30,-72,-32,-3,48,54,48,12,-38,60,56,18,-42,-96,
%U A323363 -44,-36,-24,72,-48,-12,8,-18,72,-42,-54,12,72,24,80,90,-60,72,-62,96,-32,3,84,-144,-68,-54,96,-144,-72,-12,-74,114,-24
%N A323363 Dirichlet inverse of Dedekind's psi, A001615.
%H A323363 Antti Karttunen, <a href="/A323363/b323363.txt">Table of n, a(n) for n = 1..16384</a>
%F A323363 G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} psi(k) * A(x^k). - _Ilya Gutkovskiy_, Sep 04 2019
%F A323363 From _Amiram Eldar_, Oct 14 2020: (Start)
%F A323363 Multiplicative with a(p^e) = (-1)^e * (p+1).
%F A323363 a(n) = A008836(n) * A048250(n). (End)
%F A323363 Dirichlet g.f.: zeta(2*s)/(zeta(s-1)*zeta(s)). - _Amiram Eldar_, Dec 05 2022
%t A323363 psi[n_] := If[n == 1, 1, n Times @@ (1 + 1/FactorInteger[n][[All, 1]])];
%t A323363 a[n_] := a[n] = If[n == 1, 1, -Sum[psi[n/d] a[d], {d, Most@ Divisors[n]}]];
%t A323363 Array[a, 75] (* _Jean-François Alcover_, Feb 15 2020 *)
%t A323363 f[p_, e_] := (-1)^e * (p + 1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* _Amiram Eldar_, Oct 14 2020 *)
%o A323363 (PARI)
%o A323363 A001615(n) = (n * sumdivmult(n, d, issquarefree(d)/d)); \\ From A001615
%o A323363 A323363(n) = if(1==n,1,-sumdiv(n,d,if(d<n,A001615(n/d)*A323363(d),0)));
%Y A323363 Cf. A048250 (absolute values).
%Y A323363 Cf. A001615, A008836, A323364, A323372.
%K A323363 sign,mult,easy
%O A323363 1,2
%A A323363 _Antti Karttunen_, Jan 13 2019