A353367 Sum of A110963 and its Dirichlet inverse.

2, 0, 0, 1, 0, 2, 0, 1, 1, 4, 0, 1, 0, 2, 4, 1, 0, 5, 0, 2, 2, 4, 0, 1, 4, 8, 5, 1, 0, -2, 0, 1, 4, 10, 4, 3, 0, 6, 8, 2, 0, 10, 0, 2, 8, 4, 0, 1, 1, 10, 10, 4, 0, 3, 8, 1, 6, 16, 0, 1, 0, 2, 15, 1, 16, 14, 0, 5, 4, 6, 0, 3, 0, 20, 6, 3, 4, -2, 0, 2, 9, 22, 0, 6, 20, 12, 16, 2, 0, 16, 8, 2, 2, 4, 12, 1, 0, 25, 24

Offset: 1

Author

Antti Karttunen, Apr 18 2022

Keywords

• sign

Comments

Note the negative terms, in contrast to A349135, which apparently has none.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384

Crossrefs

Cf. A003602, A110963, A353366.

Cf. also A349135, A353369.

Programs

PARI

up_to = 65537;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA003602(n) = (1+(n>>valuation(n,2)))/2;
A110963(n) = if(n%2, A003602((1+n)/2), A110963(n/2));
v353366 = DirInverseCorrect(vector(up_to,n,A110963(n)));
A353366(n) = v353366[n];
A353367(n) = (A110963(n)+A353366(n));

Formula

a(n) = A110963(n) + A353366(n).

For n > 1, a(n) = -Sum_{d|n, 1A110963(d) * A353366(n/d).

For all n >= 1, a(4*n) = A110963(n), and a(8*n-4) = A003602(n).