You're going to need some kind of result on the occurence of primes at least, because if you remove all factors greater than some prime you trivially get perfec

You're going to need some kind of result on the occurence of primes at least, because if you remove all factors greater than some prime you trivially get perfect powers (e.g. 6! ≈ 12² if you ignore the factor 5). (After thinking about it some more, perhaps not trivially.)