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A347085 Sum of A129283 and its Dirichlet inverse.

%I A347085 #9 Nov 15 2021 01:31:09
%S A347085 2,0,0,9,0,24,0,21,16,36,0,22,0,48,48,49,0,34,0,36,64,72,0,63,36,84,
%T A347085 56,50,0,-26,0,113,96,108,96,102,0,120,112,101,0,-30,0,78,76,144,0,
%U A347085 156,64,90,144,92,0,126,144,139,160,180,0,203,0,192,104,257,168,-38,0,120,192,-34,0,230,0,228,124,134,192
%N A347085 Sum of A129283 and its Dirichlet inverse.
%H A347085 Antti Karttunen, <a href="/A347085/b347085.txt">Table of n, a(n) for n = 1..16384</a>
%F A347085 a(n) = A129283(n) + A347084(n).
%F A347085 For n > 1, a(n) = -Sum_{d|n, 1<d<n} A129283(d) * A347084(n/d).
%o A347085 (PARI)
%o A347085 up_to = 16384;
%o A347085 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 A347085 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A347085 A129283(n) = (n + A003415(n));
%o A347085 v347084 = DirInverseCorrect(vector(up_to,n,A129283(n)));
%o A347085 A347084(n) = v347084[n];
%o A347085 A347085(n) = (A129283(n)+A347084(n));
%Y A347085 Cf. A003415, A129283, A347083, A347084, A347086.
%K A347085 sign
%O A347085 1,1
%A A347085 _Antti Karttunen_, Aug 17 2021