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A340367 Dirichlet inverse of sequence b(n) = 1-A318833(n).

%I A340367 #8 Jan 06 2021 09:07:27
%S A340367 -1,0,0,2,0,7,0,6,6,13,0,13,0,19,22,10,0,19,0,23,32,31,0,-3,20,37,24,
%T A340367 33,0,21,0,6,52,49,58,-36,0,55,62,-9,0,29,0,53,52,67,0,-87,42,53,82,
%U A340367 63,0,-29,94,-15,92,85,0,-219,0,91,74,-22,112,45,0,83,112,45,0,-257,0,109,82,93,136,53,0,-165,42,121
%N A340367 Dirichlet inverse of sequence b(n) = 1-A318833(n).
%H A340367 Antti Karttunen, <a href="/A340367/b340367.txt">Table of n, a(n) for n = 1..16384</a>
%F A340367 a(1) = -1, for n > 1, a(n) = Sum_{d|n, d<n} (1-A318833(n/d)) * a(d).
%o A340367 (PARI)
%o A340367 up_to = 65537;
%o A340367 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 A340367 A023900(n) = factorback(apply(p -> 1-p, factor(n)[, 1]));
%o A340367 A318833(n) = (n+A023900(n));
%o A340367 v340367 = DirInverseCorrect(vector(up_to, n, 1-A318833(n)));
%o A340367 A340367(n) = v340367[n];
%o A340367 (PARI)
%o A340367 \\ Or as:
%o A340367 A340367(n) = if(1==n, -1, sumdiv(n, d, if(d<n, (1-A318833(n/d))*A340367(d), 0)));
%Y A340367 Cf. A000010, A023900, A318833, A340198.
%Y A340367 Cf. also A340140, A340197 for similar definitions.
%K A340367 sign
%O A340367 1,4
%A A340367 _Antti Karttunen_, Jan 05 2021