Erdős Problems

Tags Prizes

OPEN

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 $n\to \infty$.

A problem of Erdős and Selfridge, who proved (see [Er82c]) \[h(n) \asymp \left(\frac{n}{\log n}\right)^{1/2}.\]