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Source. Lemma 1, p. 3, of P. Erdős, S. W. Graham, A. Ivić and C. Pomerance, On the number of divisors of n!, Analytic Number Theory (Progress in Mathematics), Birkhäuser Boston (1996), 337--355, doi:10.1007/978-1-4612-4086-0_19, read in the authors' manuscript named on the source card; pages here are that manuscript's printed pages 1--16, and the published pagination was not compared.

Statement

Lemma 1 (p. 3). "Let $S(n)$ denote the sum of the prime factors of $n$ where they are summed with multiplicity. Then for every integer $n\ge1$,

$$1+\frac{S(n)}{2n}\le\frac{d(n!)}{d((n-1)!)}\le1+\frac{2S(n)}{n}."$$

Here $d(m)$ is the number of positive divisors of $m$. The proof also records (display (5), p. 4) the upper bound

$$\frac{d(n!)}{d((n-1)!)}\le\exp\bigl(S(n)/n\bigr),$$

which the paper uses again in the proof of Corollary 2.

Read depth. Claims checked: the statement and display (5) were read clause by clause on the page images on 2026-10-08, and the proof on pp. 3--4 was followed step by step. Nothing here is independently reviewed.

Proof sketch

Pp. 3--4. Only the primes $p$ dividing $n$ change exponent from $(n-1)!$ to $n!$, so the ratio is $\prod_{p^a\Vert n}\bigl(1+a/(w_p(n-1)+1)\bigr)$, with $w_p$ the exponent of $p$ in the factorial. For $p\mid n$ one has $w_p(n-1)+1\ge n/p$, which bounds each factor by $1+ap/n$ and the product by $\exp(S(n)/n)$; since $S(n)\le n$ this is at most $1+2S(n)/n$. For the lower bound, $w_p(n-1)+1\le2n/p$ for $2\le p\le n$, and the product is at least $1+\sum a/(w_p(n-1)+1)\ge1+S(n)/(2n)$.

Dependencies

None beyond the formula for the exponent of a prime in a factorial.

Bears on

• Problem 419: the lemma is the first step of Theorem 2, from which the paper reads off the limit points the problem asks for.
• Problem 420: taking products over $n<m\le n+k$, the lemma and display (5) bound the problem's ratio $\tau((n+k)!)/\tau(n!)$ above by $\exp\bigl(\sum_{i=1}^{k}S(n+i)/(n+i)\bigr)$ and below by $\prod_{i=1}^{k}\bigl(1+S(n+i)/(2(n+i))\bigr)$; the paper's bounds on $K(n)$ in Corollaries 2 and 3 are proved this way.