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A349376 Dirichlet convolution of A006368 with the Dirichlet inverse of A006369, where A006368 is the "amusical permutation", and A006369 is its inverse permutation.

%I A349376 #8 Nov 25 2021 19:39:05
%S A349376 1,0,0,1,-3,5,-4,-2,1,11,-7,-7,-7,14,7,4,-10,2,-11,-22,10,25,-14,16,7,
%T A349376 25,0,-26,-17,-41,-18,-8,17,36,34,7,-21,39,17,52,-24,-52,-25,-48,1,50,
%U A349376 -28,-36,8,-51,24,-48,-31,7,62,60,27,61,-35,136,-35,64,0,16,62,-93,-39,-70,34,-178,-42,-26,-42,75,-27,-74
%N A349376 Dirichlet convolution of A006368 with the Dirichlet inverse of A006369, where A006368 is the "amusical permutation", and A006369 is its inverse permutation.
%C A349376 Obviously, convolving this sequence with A006369 gives its inverse A006368 from n >= 1 onward.
%H A349376 Antti Karttunen, <a href="/A349376/b349376.txt">Table of n, a(n) for n = 1..20000</a>
%F A349376 a(n) = Sum_{d|n} A006368(d) * A349368(n/d).
%o A349376 (PARI)
%o A349376 A006368(n) = ((3*n)+(n%2))\(2+((n%2)*2));
%o A349376 A006369(n) = if(!(n%3),(2/3)*n,(1/3)*if(1==(n%3),((4*n)-1),((4*n)+1)));
%o A349376 memoA349368 = Map();
%o A349376 A349368(n) = if(1==n,1,my(v); if(mapisdefined(memoA349368,n,&v), v, v = -sumdiv(n,d,if(d<n,A006369(n/d)*A349368(d),0)); mapput(memoA349368,n,v); (v)));
%o A349376 A349376(n) = sumdiv(n,d,A006368(d)*A349368(n/d));
%Y A349376 Cf. A006368, A006369, A349368, A349377 (Dirichlet inverse), A349378 (sum with it).
%Y A349376 Cf. also pairs A349613, A349614 and A349397, A349398 for similar constructions.
%K A349376 sign
%O A349376 1,5
%A A349376 _Antti Karttunen_, Nov 17 2021