Problem Statement
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 $n\to \infty$.
Prove that there exists some $c>0$ such that\[h(n) \sim c \left(\frac{n}{\log n}\right)^{1/2}\]as $n\to \infty$.
Categories:
Number Theory Factorials
Progress
A problem of Erdős and Selfridge, who proved (see [Er82c])\[h(n) \asymp \left(\frac{n}{\log n}\right)^{1/2}.\]A heuristic of Tao using the Cramér model for the primes (detailed in the comments) suggests this is true with\[c=\sqrt{2\pi}=2.506\cdots.\]Source: erdosproblems.com/912 | Last verified: January 19, 2026