@usukidoll "$n$ is a multiple of $q^2$ for some $q$" and "$n$'s prime factorization has primes which are not all distinct" are equivalent. Do you see why?
(Cont'd) Taking $90$, for example: It satisfies the first one, because it's a multiple of $3^2$. It satisfies the second one, because its prime factorization is $2\!\cdot\!3\!\cdot\!3\!\cdot\!5$, and those primes aren't all different from each other.