A353421 Dirichlet convolution of A353269 with A353348 (the Dirichlet inverse of A353350).

1, 0, 0, 1, 0, -1, 0, -1, 1, 1, 0, 0, 0, -1, -1, 1, 0, 0, 0, -1, 1, 1, 0, -1, 1, -1, -1, 0, 0, 1, 0, -1, -1, 1, -1, 1, 0, -1, 1, 1, 0, 0, 0, -1, 0, 1, 0, 1, 1, -1, -1, 0, 0, -1, 1, -1, 1, -1, 0, -2, 0, 1, -1, 1, -1, 1, 0, -1, -1, 0, 0, -1, 0, -1, 0, 0, -1, 0, 0, -2, 1, 1, 0, 2, 1, -1, 1, 1, 0, 2, 1, -1, -1, 1, -1

Offset: 1

Author

Antti Karttunen, Apr 19 2022

Keywords

• sign

Comments

Dirichlet convolution between this sequence and A353352 is A353362.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A003961, A048675, A156552, A348717, A353269, A353350, A353348.

Cf. A353422 (Dirichlet inverse).

Cf. A353352, A353362.

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(dA048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A353350(n) = (0==(A048675(n)%3));
v353348 = DirInverseCorrect(vector(up_to,n,A353350(n)));
A353348(n) = v353348[n];
A156552(n) = { my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A353269(n) = (!(A156552(n)%3));
A353421(n) = sumdiv(n,d,A353269(n/d)*A353348(d));

Formula

a(n) = Sum_{d|n} A353269(n/d) * A353348(d).

a(n) = a(A003961(n)) = a(A348717(n)), for all n >= 1.