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A346483 Sum of A005171 (characteristic function of nonprimes) and its Dirichlet inverse.

%I A346483 #19 Nov 14 2021 00:49:08
%S A346483 2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,2,0,0,
%T A346483 0,3,0,0,0,2,0,0,0,0,0,0,0,4,0,0,0,0,0,2,0,2,0,0,0,4,0,0,0,2,0,0,0,0,
%U A346483 0,0,0,6,0,0,0,0,0,0,0,4,1,0,0,4,0,0,0,2,0,4,0,0,0,0,0,3,0,0,0,3,0,0,0,2,0
%N A346483 Sum of A005171 (characteristic function of nonprimes) and its Dirichlet inverse.
%C A346483 The first negative term is a(192) = -1.
%C A346483 Positions of nonzero terms are given by A033987, except for positions n = 256, 512, 6561, 16384, 19683, 32768, 390625, 1048576, ..., at which a(n) = 0 also.
%H A346483 Antti Karttunen, <a href="/A346483/b346483.txt">Table of n, a(n) for n = 1..20000</a>
%F A346483 a(n) = A005171(n) + A346482(n).
%F A346483 For n > 1, a(n) = -Sum_{d|n, 1<d<n} A005171(d) * A346482(n/d).
%t A346483 nn = 87; b = Table[If[PrimeQ[n], 1, 0], {n, nn}]; a = 1 - b; A = Table[Table[If[Mod[n, k] == 0, a[[n/k]], 0], {k, 1, nn}], {n, 1, nn}]; B = Inverse[A]; S = A[[Range[nn]]] + B[[Range[nn]]]; S[[All, 1]]
%o A346483 (PARI)
%o A346483 up_to = 65537;
%o A346483 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.
%o A346483 A005171(n) = (1-isprime(n));
%o A346483 v346482 = DirInverseCorrect(vector(up_to,n,A005171(n)));
%o A346483 A346482(n) = v346482[n];
%o A346483 A346483(n) = (A005171(n)+A346482(n));
%Y A346483 Cf. A005171, A010051, A033987, A346482.
%K A346483 sign
%O A346483 1,1
%A A346483 _Mats Granvik_ and _Antti Karttunen_, Aug 17 2021