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A353369 Sum of A103391 ("even fractal sequence") and its Dirichlet inverse.

%I A353369 #8 Apr 18 2022 17:49:20
%S A353369 2,0,0,4,0,8,0,4,4,8,0,4,0,12,8,9,0,0,0,12,12,16,0,16,4,12,0,14,0,12,
%T A353369 0,16,16,8,12,36,0,24,12,20,0,0,0,24,4,28,0,24,9,12,8,38,0,56,16,30,
%U A353369 24,20,0,34,0,36,-8,32,12,-8,0,60,28,36,0,20,0,24,8,44,24,52,0,44,28,16,0,74,8,48,20,44,0,52
%N A353369 Sum of A103391 ("even fractal sequence") and its Dirichlet inverse.
%H A353369 Antti Karttunen, <a href="/A353369/b353369.txt">Table of n, a(n) for n = 1..16384</a>
%F A353369 a(n) = A103391(n) + A353368(n).
%F A353369 For n > 1, a(n) = -Sum_{d|n, 1<d<n} A103391(d) * A353368(n/d).
%o A353369 (PARI)
%o A353369 up_to = 65537;
%o A353369 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 A353369 A003602(n) = (n/2^valuation(n, 2)+1)/2; \\ From A003602
%o A353369 A103391(n) = if(1==n,1,(1+A003602(n-1)));
%o A353369 v353368 = DirInverseCorrect(vector(up_to,n,A103391(n)));
%o A353369 A353368(n) = v353368[n];
%o A353369 A353369(n) = (A103391(n)+A353368(n));
%Y A353369 Cf. A003602, A103391, A353368.
%Y A353369 Cf. also A349135, A353367.
%K A353369 sign
%O A353369 1,1
%A A353369 _Antti Karttunen_, Apr 18 2022