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Source. Theorem 5, p. 13, 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

Definitions (pp. 12--13). $d(m)$ is the number of positive divisors of $m$ and $D(n)=d(n!)-d((n-1)!)$, the number of divisors of $n!$ that do not divide $(n-1)!$. A natural number $n$ is a champ if $D(n)>D(m)$ for all natural numbers $m<n$, by analogy with Ramanujan's highly composite numbers.

Theorem 5 (p. 13). "For each prime $p$, both $p$ and $2p$ are champs."

What the paper adds around it (pp. 13--14), outside the theorem: the least champ of neither form is $8$; a computation by Marc Deléglise of all champs up to $500$ found $30$ of neither form, each of the form $mp$ with $p$ a prime at least $P(m)$ and $m\in\{3,4,5,6,7\}$; the authors conjecture that there are infinitely many champs of neither form and say that this follows from the prime $k$-tuples conjecture, stating without proof, as an example they call relatively easy to show, that $3r$ is a champ whenever $q$ and $r$ are primes with $2q+1=3r$. These are reported remarks, not results of the paper.

Read depth. Claims checked: the definitions and the statement were read clause by clause on the page images on 2026-10-08, and the proof on p. 13 was followed step by step. Nothing here is independently reviewed.

Proof sketch

P. 13. For a prime $p$, every divisor of $(p-1)!$ times $p$ is a new divisor of $p!$, so $D(p)=d((p-1)!)\ge d(m!)>D(m)$ for $m<p$. For any $m$, the map $d\mapsto d/m$ sends divisors of $m!$ not dividing $(m-1)!$ injectively to divisors that do, so $D(m)\le d(m!)/2$. For an odd prime $p$, the factor $2p$ raises the exponent of $p$ from $1$ to $2$, so $d((2p)!)>\frac32d((2p-1)!)$ and $D(2p)>\frac12d((2p-1)!)\ge D(m)$ for $m<2p$; the case $2p=4$ is checked directly.

Dependencies

None beyond the divisor function's multiplicativity.

Bears on

No problem page in the corpus concerns the champs of $D(n)$.