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Statement

With $h(n)$ the number of distinct exponents in the prime factorization of $n!$, as in Theorem 1, Erdős writes on p. 27 that there is "no doubt" that some constant $c>0$ satisfies

$$h(n)=(c+o(1))\Bigl(\frac{n}{\log n}\Bigr)^{1/2}. \tag{4}$$

He adds that a proof of (4) seems to present very serious difficulties, because not enough is known about the differences of consecutive primes.

The passage is an expectation, not a result: the paper gives no proof and no candidate value of $c$.

Source. P. Erdős, Miscellaneous problems in number theory, Proceedings of the Eleventh Manitoba Conference on Numerical Mathematics and Computing (Winnipeg, Man., 1981), Congr. Numer. 34 (1982), 25--45; display (4) on p. 27. The edition read is identified on the source card.

Read depth. Claims checked: the passage was read clause by clause on the page image. There is no proof to check.

Dependencies

Theorem 1 of the same paper, which gives the order of magnitude.

Bears on

• Problem 912: display (4) is the problem's asymptotic $h(n)\sim c(n/\log n)^{1/2}$ with $c>0$, posed here as an expectation; the paper records no result on it beyond Theorem 1.