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A349377 Dirichlet convolution of A006369 with the Dirichlet inverse of A006368, where A006368 is the "amusical permutation", and A006369 is its inverse permutation.

%I A349377 #8 Nov 25 2021 17:32:44
%S A349377 1,0,0,-1,3,-5,4,2,-1,-11,7,7,7,-14,-7,-3,10,-2,11,16,-10,-25,14,-6,2,
%T A349377 -25,0,18,17,11,18,4,-17,-36,-10,20,21,-39,-17,-18,24,12,25,34,-7,-50,
%U A349377 28,2,8,-15,-24,34,31,3,-20,-16,-27,-61,35,30,35,-64,-8,-5,-20,23,39,50,-34,6,42,-44,42,-75,-15,52,-22,23
%N A349377 Dirichlet convolution of A006369 with the Dirichlet inverse of A006368, where A006368 is the "amusical permutation", and A006369 is its inverse permutation.
%C A349377 Obviously, convolving this sequence with A006368 gives its inverse A006369 from n >= 1 onward.
%H A349377 Antti Karttunen, <a href="/A349377/b349377.txt">Table of n, a(n) for n = 1..20000</a>
%F A349377 a(n) = Sum_{d|n} A006369(d) * A349351(n/d).
%F A349377 a(n) = A349378(n) - A349376(n).
%o A349377 (PARI)
%o A349377 A006368(n) = ((3*n)+(n%2))\(2+((n%2)*2));
%o A349377 A006369(n) = if(!(n%3),(2/3)*n,(1/3)*if(1==(n%3),((4*n)-1),((4*n)+1)));
%o A349377 memoA349351 = Map();
%o A349377 A349351(n) = if(1==n,1,my(v); if(mapisdefined(memoA349351,n,&v), v, v = -sumdiv(n,d,if(d<n,A006368(n/d)*A349351(d),0)); mapput(memoA349351,n,v); (v)));
%o A349377 A349377(n) = sumdiv(n,d,A006369(d)*A349351(n/d));
%Y A349377 Cf. A006368, A006369, A349351, A349376 (Dirichlet inverse), A349378 (sum with it).
%Y A349377 Cf. also pairs A349613, A349614 and A349397, A349398 for similar constructions.
%K A349377 sign
%O A349377 1,5
%A A349377 _Antti Karttunen_, Nov 17 2021