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A340094 Dirichlet inverse of A062830, n - phi(n) + 1.

%I A340094 #15 Jan 07 2021 10:41:17
%S A340094 1,-2,-2,1,-2,3,-2,-1,0,1,-2,-1,-2,-1,0,0,-2,-1,-2,3,-2,-5,-2,1,-2,-7,
%T A340094 -2,7,-2,9,-2,-1,-6,-11,-4,4,-2,-13,-8,-3,-2,17,-2,15,2,-17,-2,1,-4,
%U A340094 -3,-12,19,-2,3,-8,-7,-14,-23,-2,-11,-2,-25,4,-1,-10,33,-2,27,-18,17,-2,-3,-2,-31,-4,31,-10,41,-2,5
%N A340094 Dirichlet inverse of A062830, n - phi(n) + 1.
%H A340094 Antti Karttunen, <a href="/A340094/b340094.txt">Table of n, a(n) for n = 1..16384</a>
%F A340094 a(1) = 1, for n > 1, a(n) = -Sum_{d|n, d<n} A062830(n/d) * a(d).
%o A340094 (PARI)
%o A340094 up_to = 16384;
%o A340094 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 A340094 A062830(n) = 1+(n-eulerphi(n));
%o A340094 v340094 = DirInverseCorrect(vector(up_to, n, A062830(n)));
%o A340094 A340094(n) = v340094[n];
%o A340094 \\ Or as:
%o A340094 A340094(n) = if(1==n, 1, -sumdiv(n, d, if(d<n, A062830(n/d)*A340094(d), 0)));
%Y A340094 Cf. A000010, A051953, A062830, A340090.
%K A340094 sign
%O A340094 1,2
%A A340094 _Antti Karttunen_, Jan 05 2021