SOLVED
Let $t(n)$ be maximal such that there is a representation
\[n!=a_1\cdots a_n\]
with $t(n)=a_1\leq \cdots \leq a_n$. Obtain good bounds for $t(n)/n$. In particular, is it true that
\[\lim \frac{t(n)}{n}=\frac{1}{e}?\]
Furthermore, does there exist some constant $c>0$ such that
\[\frac{t(n)}{n} \leq \frac{1}{e}-\frac{c}{\log n}\]
for infinitely many $n$?
It is easy to see that
\[\lim \frac{t(n)}{n}\leq \frac{1}{e}.\]
Erdős
[Er96b] wrote he, Selfridge, and Straus had proved a corresponding lower bound, so that $\lim \frac{t(n)}{n}=\frac{1}{e}$, and 'believed that Straus had written up our proof. Unfortunately Straus suddenly died and no trace was ever found of his notes. Furthermore, we never could reconstruct our proof, so our assertion now can be called only a conjecture.'
Alladi and Grinstead [AlGr77] have obtained similar results when the $a_i$ are restricted to prime powers.
Both questions were answered by Tao [Ta25], who proved that
\[\frac{1}{e}-O\left(\frac{1}{\log n}\right)\leq \frac{t(n)}{n}\leq \frac{1}{e}-\frac{c_0}{\log n}+o\left(\frac{1}{\log n}\right),\]
where $c_0=0.3044\cdots$ is an explicit constant. Tao conjectures that the upper bound is the truth, in that in fact
\[\frac{t(n)}{n}= \frac{1}{e}-\frac{c_0}{\log n}+o\left(\frac{1}{\log n}\right).\]
