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A353367 Sum of A110963 and its Dirichlet inverse.

%I A353367 #11 Apr 18 2022 17:49:15
%S A353367 2,0,0,1,0,2,0,1,1,4,0,1,0,2,4,1,0,5,0,2,2,4,0,1,4,8,5,1,0,-2,0,1,4,
%T A353367 10,4,3,0,6,8,2,0,10,0,2,8,4,0,1,1,10,10,4,0,3,8,1,6,16,0,1,0,2,15,1,
%U A353367 16,14,0,5,4,6,0,3,0,20,6,3,4,-2,0,2,9,22,0,6,20,12,16,2,0,16,8,2,2,4,12,1,0,25,24
%N A353367 Sum of A110963 and its Dirichlet inverse.
%C A353367 Note the negative terms, in contrast to A349135, which apparently has none.
%H A353367 Antti Karttunen, <a href="/A353367/b353367.txt">Table of n, a(n) for n = 1..16384</a>
%F A353367 a(n) = A110963(n) + A353366(n).
%F A353367 For n > 1, a(n) = -Sum_{d|n, 1<d<n} A110963(d) * A353366(n/d).
%F A353367 For all n >= 1, a(4*n) = A110963(n), and a(8*n-4) = A003602(n).
%o A353367 (PARI)
%o A353367 up_to = 65537;
%o A353367 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 A353367 A003602(n) = (1+(n>>valuation(n,2)))/2;
%o A353367 A110963(n) = if(n%2, A003602((1+n)/2), A110963(n/2));
%o A353367 v353366 = DirInverseCorrect(vector(up_to,n,A110963(n)));
%o A353367 A353366(n) = v353366[n];
%o A353367 A353367(n) = (A110963(n)+A353366(n));
%Y A353367 Cf. A003602, A110963, A353366.
%Y A353367 Cf. also A349135, A353369.
%K A353367 sign
%O A353367 1,1
%A A353367 _Antti Karttunen_, Apr 18 2022