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%I #11 Nov 21 2021 10:16:27
%S 1,-2,-2,0,-3,4,-4,0,-1,6,-6,0,-7,8,4,0,-9,2,-10,0,5,12,-12,0,-4,14,
%T -2,0,-15,-8,-16,0,7,18,6,0,-19,20,8,0,-21,-10,-22,0,3,24,-24,0,-9,8,
%U 10,0,-27,4,8,0,11,30,-30,0,-31,32,4,0,9,-14,-34,0,13,-12,-36,0,-37,38,8,0,9,-16,-40,0,-4,42,-42,0
%N Dirichlet inverse of A109168, where A109168(n) = (n+A006519(n))/2, and A006519 is the highest power of 2 dividing n.
%H Antti Karttunen, <a href="/A349344/b349344.txt">Table of n, a(n) for n = 1..20000</a>
%F a(1) = 1; a(n) = -Sum_{d|n, d < n} A109168(n/d) * a(d).
%F a(n) = A349345(n) - A109168(n).
%o (PARI)
%o up_to = 20000;
%o 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 A109168(n) = ((n+bitand(n, -n))\2); \\ From A109168 by _M. F. Hasler_, Oct 19 2019 (Cf. A140472).
%o v349344 = DirInverseCorrect(vector(up_to,n,A109168(n)));
%o A349344(n) = v349344[n];
%Y Cf. A109168 (A140472), A349345.
%Y Cf. also A328203, A349134, A349343, A349346, A349353.
%K sign
%O 1,2
%A _Antti Karttunen_, Nov 15 2021
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