Erdős Problem #912

declared status 'open'. Formalized: yes. 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 $n\to \infty$. Prize: no. OEIS: A071626. Tags: factorials, number theory.

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