A353338 Number of ways to write the square of n as a product of the terms of A028260 larger than 1; a(1) = 1 by convention (an empty product).

1, 1, 1, 2, 1, 3, 1, 3, 2, 3, 1, 6, 1, 3, 3, 5, 1, 6, 1, 6, 3, 3, 1, 12, 2, 3, 3, 6, 1, 12, 1, 7, 3, 3, 3, 16, 1, 3, 3, 12, 1, 12, 1, 6, 6, 3, 1, 21, 2, 6, 3, 6, 1, 12, 3, 12, 3, 3, 1, 33, 1, 3, 6, 11, 3, 12, 1, 6, 3, 12, 1, 33, 1, 3, 6, 6, 3, 12, 1, 21, 5, 3, 1, 33, 3, 3, 3, 12, 1, 33, 3, 6, 3, 3, 3, 36, 1, 6, 6

Offset: 1

Author

Antti Karttunen, Apr 17 2022

Keywords

• nonn

Comments

Number of factorizations of n^2 into factors k > 1 for which there is an even number of primes (when counted with multiplicity, A001222) in their prime factorization.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Index entries for sequences computed from exponents in factorization of n

Crossrefs

Cf. A000290, A001222, A028260, A038548, A046523, A065043, A353337.

Cf. also A353378.

Programs

PARI

A065043(n) = (1 - (bigomega(n)%2));
A353337(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m)&&A065043(d), s += A353337(n/d, d))); (s));
A353338(n) = A353337(n^2);

Formula

a(n) = A353337(A000290(n)).

a(n) = a(A046523(n)). [The sequence depends only on the prime signature of n].

For all n >= 1, a(n) >= A353378(n).