A346242 Dirichlet inverse of A324198, where A324198(n) = gcd(n, A276086(n)).

1, -1, -3, 0, -1, 5, -1, 0, 6, -3, -1, -2, -1, 1, -9, 0, -1, -16, -1, 4, 3, 1, -1, 0, -24, 1, -12, 0, -1, 43, -1, 0, 3, 1, -5, 14, -1, 1, 3, 0, -1, -11, -1, 0, 54, 1, -1, 0, -6, 32, 3, 0, -1, 44, -3, -6, 3, 1, -1, -50, -1, 1, -24, 0, 1, -5, -1, 0, 3, -15, -1, -4, -1, 1, 96, 0, -5, -5, -1, 0, 24, 1, -1, 8, -3, 1, 3, 0, -1

Offset: 1

Author

Antti Karttunen, Jul 13 2021

Keywords

• sign
• base

Links

• Antti Karttunen, Table of n, a(n) for n = 1..30030
• Index entries for sequences related to primorial base

Crossrefs

Cf. A276086, A324198, A346243.

Cf. A008966 (parity of terms), A005117 (positions of odd terms), A013929 (of even terms), A045344 (of -1's, at least a subset of them), A354810 (of 0's), A354811 (of 1's), A354812 (of 2's), A354813 (of 3's), A354814 (of 4's), A354822 (of -2's).

Cf. also A354347, A354348, A354823, A354824.

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(dA324198(n) = { my(m=1, p=2, orgn=n); while(n, m *= (p^min(n%p, valuation(orgn, p))); n = n\p; p = nextprime(1+p)); (m); };
v346242 = DirInverseCorrect(vector(up_to,n,A324198(n)));
A346242(n) = v346242[n];

Formula

a(n) = A346243(n) - A324198(n).

From Antti Karttunen, Jun 09 2022: (Start)

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA324198(n/d) * a(d).

For all n >= 1, A000035(a(n)) = A008966(n).

For all n >= 1, a(A045344(n)) = -1.

(End)