cp's OEIS Frontend

A340090 Dirichlet inverse of A219428, n - phi(n) - 1.

%I A340090 #16 Jan 07 2021 10:41:00
%S A340090 -1,0,0,-1,0,-3,0,-3,-2,-5,0,-7,0,-7,-6,-8,0,-11,0,-11,-8,-11,0,-21,
%T A340090 -4,-13,-8,-15,0,-21,0,-21,-12,-17,-10,-36,0,-19,-14,-33,0,-29,0,-23,
%U A340090 -20,-23,0,-63,-6,-29,-18,-27,0,-47,-14,-45,-20,-29,0,-85,0,-31,-26,-55,-16,-45,0,-35,-24,-45,0,-123,0,-37
%N A340090 Dirichlet inverse of A219428, n - phi(n) - 1.
%H A340090 Antti Karttunen, <a href="/A340090/b340090.txt">Table of n, a(n) for n = 1..16384</a>
%F A340090 a(1) = -1, for n > 1, a(n) = Sum_{d|n, d<n} A219428(n/d) * a(d).
%o A340090 (PARI)
%o A340090 up_to = 2^14;
%o A340090 DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(d<n, v[n/d]*u[d], 0)))); (u) }; \\ Compute the Dirichlet inverse of the sequence given in input vector v (correctly!).
%o A340090 A219428(n) = (n - 1 - eulerphi(n));
%o A340090 v340090 = DirInverseCorrect(vector(up_to, n, A219428(n)));
%o A340090 A340090(n) = v340090[n];
%o A340090 \\ Or as:
%o A340090 A340090(n) = if(1==n, -1, sumdiv(n, d, if(d<n, A219428(n/d)*A340090(d), 0)));
%Y A340090 Cf. A000010, A033987, A051953, A219428, A340094, A340140, A340197, A340198.
%K A340090 sign
%O A340090 1,6
%A A340090 _Antti Karttunen_, Jan 05 2021