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A349351 Dirichlet inverse of A006368, "the amusical permutation", a(2n)=3n, a(4n+1)=3n+1, a(4n-1)=3n-1.

%I A349351 #8 Nov 25 2021 17:36:11
%S A349351 1,-3,-2,3,-4,3,-5,-3,-3,9,-8,6,-10,9,5,3,-13,15,-14,0,4,15,-17,-15,
%T A349351 -3,21,0,9,-22,9,-23,-3,7,27,14,-24,-28,27,11,-9,-31,27,-32,12,18,33,
%U A349351 -35,24,-12,15,14,6,-40,9,23,-27,13,45,-44,-63,-46,45,27,3,31,39,-50,9,16,9,-53,6,-55,57,12,18,22,33,-59
%N A349351 Dirichlet inverse of A006368, "the amusical permutation", a(2n)=3n, a(4n+1)=3n+1, a(4n-1)=3n-1.
%H A349351 Antti Karttunen, <a href="/A349351/b349351.txt">Table of n, a(n) for n = 1..20000</a>
%F A349351 a(1) = 1; a(n) = -Sum_{d|n, d < n} A006368(n/d) * a(d).
%F A349351 a(n) = A349352(n) - A006368(n).
%o A349351 (PARI)
%o A349351 up_to = 20000;
%o A349351 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 A349351 A006368(n) = ((3*n)+(n%2))\(2+((n%2)*2));
%o A349351 v349351 = DirInverseCorrect(vector(up_to,n,A006368(n)));
%o A349351 A349351(n) = v349351[n];
%Y A349351 Cf. A006368, A349352, A349368, A349377.
%K A349351 sign
%O A349351 1,2
%A A349351 _Antti Karttunen_, Nov 15 2021