A373370 a(n) = gcd(bigomega(n), A064097(n)), where bigomega is number of prime factors with repetition, and A064097 is a quasi-logarithm defined inductively by a(1) = 0 and a(p) = 1 + a(p-1) if p is prime and a(n*m) = a(n) + a(m) if m,n > 1.

0, 1, 1, 2, 1, 1, 1, 3, 2, 2, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 3, 3, 1, 3, 1, 5, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 2, 1, 6, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 4, 2, 1, 4, 2, 1, 1, 4, 1, 4, 1, 3, 1, 2, 1, 1, 1, 3, 3, 4, 1, 1, 1, 4, 3

Offset: 1

Author

Antti Karttunen, Jun 05 2024

Keywords

• nonn

Comments

As A001222 and A064097 are both fully additive sequences, all sequences that give the positions of multiples of some natural number k in this sequence are closed under multiplication, i.e., are multiplicative semigroups.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..100000

Crossrefs

Cf. A001222, A064097.

Cf. also A373365, A373366.

Programs

PARI

A064097(n) = if(1==n,0,1+A064097(n-(n/vecmin(factor(n)[,1]))));
A373370(n) = gcd(bigomega(n), A064097(n));