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A347086 Difference between the Dirichlet inverse of -A168036, n - A003415(n) and the Dirichlet inverse of A129283, n + A003415(n), where A003415 is the Arithmetic derivative of n.

%I A347086 #8 Nov 15 2021 01:31:34
%S A347086 0,2,2,0,2,-10,2,2,0,-14,2,6,2,-18,-16,8,2,6,2,6,-20,-26,2,6,0,-30,2,
%T A347086 6,2,74,2,26,-28,-38,-24,0,2,-42,-32,2,2,94,2,6,6,-50,2,10,0,6,-40,6,
%U A347086 2,14,-32,-2,-44,-62,2,-48,2,-66,6,80,-36,134,2,6,-52,130,2,20,2,-78,6,6,-36,154,2,-6,12,-86,2,-60
%N A347086 Difference between the Dirichlet inverse of -A168036, n - A003415(n) and the Dirichlet inverse of A129283, n + A003415(n), where A003415 is the Arithmetic derivative of n.
%H A347086 Antti Karttunen, <a href="/A347086/b347086.txt">Table of n, a(n) for n = 1..16384</a>
%F A347086 a(n) = A347082(n) - A347084(n).
%o A347086 (PARI)
%o A347086 up_to = 16384;
%o A347086 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 A347086 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A347086 v347082 = DirInverseCorrect(vector(up_to,n,n-A003415(n)));
%o A347086 A347082(n) = v347082[n];
%o A347086 v347084 = DirInverseCorrect(vector(up_to,n,n+A003415(n)));
%o A347086 A347084(n) = v347084[n];
%o A347086 A347086(n) = (A347082(n)-A347084(n));
%Y A347086 Cf. A003415, A129283, A168036, A347082, A347084.
%K A347086 sign
%O A347086 1,2
%A A347086 _Antti Karttunen_, Aug 17 2021