Erdős Problem #912 — If $n! = \prod_i p_i^{k_i}$ is the factorisation into distinct primes then let $h(n)$ count the number of distinct exponents $k_i$.

If $n! = \prod_i p_i^{k_i}$ is the factorisation into distinct primes then let $h(n)$ count the number of distinct exponents $k_i$. Prove that there exists some $c>0$ such that $h(n) \sim c \left(\frac{n}{\log n}\right)^{1/2}$ as $n\to \infty$.