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A347083 Sum of -A168036 and its Dirichlet inverse.

%I A347083 #8 Nov 15 2021 01:31:46
%S A347083 2,0,0,1,0,4,0,-1,4,8,0,-4,0,12,16,-7,0,-2,0,-6,24,20,0,-19,16,24,4,
%T A347083 -8,0,-14,0,-21,40,32,48,-18,0,36,48,-33,0,-18,0,-12,4,44,0,-58,36,6,
%U A347083 64,-14,0,-22,80,-47,72,56,0,-29,0,60,8,-47,96,-26,0,-18,88,-22,0,-62,0,72,20,-20,120,-30,0,-108,-11,80
%N A347083 Sum of -A168036 and its Dirichlet inverse.
%H A347083 Antti Karttunen, <a href="/A347083/b347083.txt">Table of n, a(n) for n = 1..16384</a>
%F A347083 a(n) = A347082(n) - A168036(n).
%F A347083 For n > 1, a(n) = Sum_{d|n, 1<d<n} A168036(d) * A347082(n/d).
%o A347083 (PARI)
%o A347083 up_to = 16384;
%o A347083 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 A347083 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A347083 A168036(n) = (A003415(n)-n);
%o A347083 v347082 = DirInverseCorrect(vector(up_to,n,-A168036(n)));
%o A347083 A347082(n) = v347082[n];
%o A347083 A347083(n) = (A347082(n)-A168036(n));
%Y A347083 Cf. A003415, A168036, A347082, A347085.
%K A347083 sign
%O A347083 1,1
%A A347083 _Antti Karttunen_, Aug 17 2021