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

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, 25, 0, -26, -17, -41, -18, -8, 17, 36, 34, 7, -21, 39, 17, 52, -24, -52, -25, -48, 1, 50, -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

Offset: 1

Author

Antti Karttunen, Nov 17 2021

Keywords

• sign

Comments

Obviously, convolving this sequence with A006369 gives its inverse A006368 from n >= 1 onward.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A006368, A006369, A349368, A349377 (Dirichlet inverse), A349378 (sum with it).

Cf. also pairs A349613, A349614 and A349397, A349398 for similar constructions.

Programs

PARI

A006368(n) = ((3*n)+(n%2))\(2+((n%2)*2));
A006369(n) = if(!(n%3),(2/3)*n,(1/3)*if(1==(n%3),((4*n)-1),((4*n)+1)));
memoA349368 = Map();
A349368(n) = if(1==n,1,my(v); if(mapisdefined(memoA349368,n,&v), v, v = -sumdiv(n,d,if(dA006369(n/d)*A349368(d),0)); mapput(memoA349368,n,v); (v)));
A349376(n) = sumdiv(n,d,A006368(d)*A349368(n/d));

Formula

a(n) = Sum_{d|n} A006368(d) * A349368(n/d).