A353419 Sum of A353269 and its Dirichlet inverse.

2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2

Offset

1,1

Comments

The first negative term is a(840) = -2.

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for sequences computed from indices in prime factorization

Formula

a(n) = A353269(n) + A353418(n).

For n > 1, a(n) = -Sum_{d|n, 1<d<n} A353269(d) * A353418(n/d).

a(p) = 0 for all primes p.

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

Prog

(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(d<n, v[n/d]*u[d], 0)))); (u) }; \\ Compute the Dirichlet inverse of the sequence given in input vector v
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));
v353418 = DirInverseCorrect(vector(up_to, n, A353269(n)));
A353418(n) = v353418[n];
A353419(n) = (A353269(n)+A353418(n));

Crossrefs

Cf. A003961, A156552, A348717, A353269, A353418.

Cf. also A353349.

Sequence in context: A112315 A295663 A005926 * A346483 A349378 A331918

Adjacent sequences: A353416 A353417 A353418 * A353420 A353421 A353422

Keyword

sign

Author

Antti Karttunen, Apr 19 2022

Status

approved