A353337 Number of ways to write n as a product of the terms of A028260 larger than 1; a(1) = 1 by convention (an empty product).

1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 2, 0, 0, 0, 0, 1, 1, 0, 2, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 3, 0, 1, 1, 2, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 2, 1, 2, 1, 1, 0, 3, 0, 1, 0, 3, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 2, 1, 0, 3, 1, 1, 1, 2, 0, 3, 1, 0, 1, 1, 1, 4, 0, 0, 0, 3, 0, 0, 0, 2, 0

Offset: 1

Author

Antti Karttunen, Apr 17 2022

Keywords

• nonn

Comments

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

Examples

			Of the eleven divisors of 96 larger than one, the following: [4, 6, 16, 24, 96] are terms of A028260 because they have an even number of prime factors when counted with repetition. Using them, we can factor 96 in four possible ways, as 96 = 24*4 = 16*6 = 6*4*4, therefore a(96) = 4.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences computed from exponents in factorization of n

Crossrefs

Cf. A001222, A028260, A038548, A046523, A065043, A353338 [= a(n^2)].

Cf. also A320655, A353377.

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));

Formula

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

For all n >= 1, a(n) >= A320655(n), and a(n) >= A353377(n).