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Source. Theorem 3, p. 5, 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

Theorem 3 (p. 5). "Recall that $S(n)$ denotes the sum of the prime factors of $n$, with multiplicity. Let $f(n)$ denote the least number such that

$$\sum_{i=1}^{f(n)}S(n+i)>n.$$

For each number $\varepsilon>0$ there are infinitely many integers $n$ for which

$$f(n)\ge(1/4-\varepsilon)\log n\log\log n\log\log\log\log n/(\log\log\log n)^3."$$

Read depth. Claims checked: the statement was read clause by clause on the page image on 2026-10-08; the proof on pp. 6--8, including Lemma 2 (p. 7), was read for structure only. Nothing here is independently reviewed.

Proof sketch

Pp. 6--8. For a large parameter $u$ let $M$ be the product of the primes in $[\log^2u,u]$. The Erdős--Rankin construction, with de Bruijn's count of smooth numbers and Mertens' theorem, gives a residue class $A$ modulo $M$ such that each of $A+1,\dots,A+L$ shares a prime factor with $M$, where $L=(1/2-\varepsilon/4)u\log u\log\log\log u/(\log\log u)^3$. For $j$ in $[M/2,M]$ the numbers $jM+A+i$ ($1\le i\le L$) are of size about $M^2$, and their prime factors above $(jM+A+i)/u^3$ are controlled by a sieve bound (the paper's Lemma 2, p. 7) on how often $(jM+A+i)/l$ is prime. Summing over $j$ shows that the double sum of $S(jM+A+i)$ over $M/2\le j\le M$ and $1\le i\le L$ is $o(M^3)$, so some $j$ has $\sum_{i\le L}S(jM+A+i)<jM+A$, that is $f(jM+A)>L$; since $\log(jM+A)$ is about $2u$, this is the stated bound.

Dependencies

The Erdős--Rankin method, de Bruijn's estimate for smooth numbers, Mertens' theorem, a sieve upper bound for primes in progressions, and the paper's Lemma 2 (p. 7), which is not given its own page here.

Bears on

• Problem 420: through Corollary 2, which the paper derives from this theorem; see that page for the relation.