Problem 912

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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 $n\to \infty$.

Status. Open.

Source. erdosproblems.com/912, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #912, https://www.erdosproblems.com/912.

References.

• [Er82c] Erdős, P., Miscellaneous problems in number theory. Congr. Numer. (1982), 25-45.

Formalization. Statement in formal-conjectures.

Progress

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Known Results

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Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• erdos_1982_miscellaneous_problems_number_theory
• erdos_1982_miscellaneous_problems_number_theory / display_4
• erdos_1982_miscellaneous_problems_number_theory / theorem_1