A353418 Dirichlet inverse of A353269.

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

Offset: 1

Author

Antti Karttunen, Apr 19 2022

Keywords

• sign

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, A156552, A348717, A353269, A353419.

Cf. also A353348, A353422.

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(dA156552(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));
v353418 = DirInverseCorrect(vector(up_to,n,A353269(n)));
A353418(n) = v353418[n];

Formula

a(1) = 1; a(n) = -Sum_{d|n, d < n} A353269(n/d) * a(d).

a(n) = A353419(n) - A353269(n).

a(p) = 0 for all primes p.

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