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A353461 Dirichlet convolution of A003602 (Kimberling's paraphrases) with A323881 (the Dirichlet inverse of A126760).

%I A353461 #15 Jul 08 2022 21:40:18
%S A353461 1,0,1,0,1,0,1,0,3,0,2,0,2,0,3,0,3,0,3,0,4,0,4,0,2,0,9,0,5,0,5,0,7,0,
%T A353461 1,0,6,0,8,0,7,0,7,0,9,0,8,0,5,0,11,0,9,0,1,0,12,0,10,0,10,0,12,0,2,0,
%U A353461 11,0,15,0,12,0,12,0,10,0,3,0,13,0,27,0,14,0,2,0,19,0,15,0,4,0,20,0,3,0,16,0,21
%N A353461 Dirichlet convolution of A003602 (Kimberling's paraphrases) with A323881 (the Dirichlet inverse of A126760).
%C A353461 Taking the Dirichlet convolution between this sequence and A349393 gives A349371, and similarly for many other such analogous pairs.
%H A353461 Antti Karttunen, <a href="/A353461/b353461.txt">Table of n, a(n) for n = 1..65537</a>
%F A353461 a(n) = Sum_{d|n} A003602(d) * A323881(n/d).
%F A353461 a(n) = A353462(n) - A353460(n).
%o A353461 (PARI)
%o A353461 up_to = 65537;
%o A353461 A003602(n) = (1+(n>>valuation(n,2)))/2;
%o A353461 A126760(n) = {n&&n\=3^valuation(n, 3)<<valuation(n, 2); n%3+n\6*2}; \\ From A126760
%o A353461 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 A353461 v323881 = DirInverseCorrect(vector(up_to,n,A126760(n)));
%o A353461 A323881(n) = v323881[n];
%o A353461 A353461(n) = sumdiv(n,d,A003602(d)*A323881(n/d));
%Y A353461 Cf. A003602, A126760, A323881, A353460 (Dirichlet inverse), A353462 (sum with it).
%Y A353461 Cf. also A349371, A349393.
%K A353461 sign
%O A353461 1,9
%A A353461 _Antti Karttunen_, Apr 20 2022