A373381 a(n) = gcd(bigomega(n), A056239(n)), where bigomega is number of prime factors with repetition, and A056239 is fully additive with a(p) = primepi(p).

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

Offset: 1

Author

Antti Karttunen, Jun 06 2024

Keywords

• nonn

Comments

As A001222 and A056239 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; for example A340784.

Links

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

Crossrefs

Cf. A001222, A056239, A340784 (positions of even terms), A353331 (their characteristic function).

Cf. also A354871, A373370.

Programs

PARI

A056239(n) = if(1==n, 0, my(f=factor(n)); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1])));
A373381(n) = gcd(bigomega(n), A056239(n));

Formula

a(n) = gcd(A001222(n), A056239(n)).