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A323882 Sum of A126760 and its Dirichlet inverse.

%I A323882 #15 Aug 18 2021 21:43:33
%S A323882 2,0,0,1,0,2,0,1,1,4,0,1,0,6,4,1,0,1,0,2,6,8,0,1,4,10,1,3,0,0,0,1,8,
%T A323882 12,12,1,0,14,10,2,0,0,0,4,2,16,0,1,9,14,12,5,0,1,16,3,14,20,0,2,0,22,
%U A323882 3,1,20,0,0,6,16,12,0,1,0,26,14,7,24,0,0,2,1,28,0,3,24,30,20,4,0,2,30,8,22,32,28,1,0,25,4,9,0,0,0,5,12
%N A323882 Sum of A126760 and its Dirichlet inverse.
%C A323882 From _Antti Karttunen_, Aug 18 2021: (Start)
%C A323882 No negative terms in range 1 .. 2^20.
%C A323882 Apparently zeros occur only on (some of the) positions given by A030059, with exceptions for example on n = 70, 105, 110, 130, 154, etc, where a(n) > 0.
%C A323882 (End)
%H A323882 Antti Karttunen, <a href="/A323882/b323882.txt">Table of n, a(n) for n = 1..20000</a>
%H A323882 Antti Karttunen, <a href="/A323882/a323882.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%F A323882 a(n) = A126760(n) + A323881(n).
%F A323882 For n > 1, a(n) = -Sum_{d|n, 1<d<n} A126760(d) * A323881(n/d). - _Antti Karttunen_, Aug 18 2021
%o A323882 (PARI)
%o A323882 up_to = 20000;
%o A323882 A126760(n) = {n&&n\=3^valuation(n, 3)<<valuation(n, 2); n%3+n\6*2}; \\ From A126760
%o A323882 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 A323882 v323881 = DirInverseCorrect(vector(up_to,n,A126760(n)));
%o A323882 A323881(n) = v323881[n];
%o A323882 A323882(n) = (A126760(n)+A323881(n));
%Y A323882 Cf. A030059, A126760, A323881, A323884, A323885, A323887.
%K A323882 nonn,look
%O A323882 1,1
%A A323882 _Antti Karttunen_, Feb 08 2019