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A347082 Dirichlet inverse of -A168036, n - A003415(n).

%I A347082 #14 Nov 15 2021 01:32:25
%S A347082 1,-1,-2,1,-4,3,-6,3,1,5,-10,0,-12,7,9,9,-16,1,-18,-2,13,11,-22,1,1,
%T A347082 13,4,-4,-28,-13,-30,27,21,17,25,6,-36,19,25,-5,-40,-19,-42,-8,-2,23,
%U A347082 -46,6,1,1,33,-10,-52,5,41,-11,37,29,-58,3,-60,31,-4,81,49,-31,-66,-14,45,-33,-70,22,-72,37,0,-16,61,-37,-78
%N A347082 Dirichlet inverse of -A168036, n - A003415(n).
%H A347082 Antti Karttunen, <a href="/A347082/b347082.txt">Table of n, a(n) for n = 1..16384</a>
%F A347082 a(1) = 1; and for n > 2, a(n) = Sum_{d|n, d<n} a(d) * A168036(n/d).
%F A347082 a(n) = A168036(n) + A347083(n).
%F A347082 a(n) = A347084(n) + A347086(n).
%F A347082 a(n) = A347087(n) - n.
%o A347082 (PARI)
%o A347082 up_to = 16384;
%o A347082 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 A347082 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A347082 v347082 = DirInverseCorrect(vector(up_to,n,n-A003415(n)));
%o A347082 A347082(n) = v347082[n];
%Y A347082 Cf. A003415, A168036, A347083, A347084, A347086, A347087.
%Y A347082 Cf. also A346241.
%K A347082 sign
%O A347082 1,3
%A A347082 _Antti Karttunen_, Aug 17 2021