Is there some function $f(r)$ such that $f(r)\to \infty$ as $r\to\infty$, such that, for infinitely many $n$, there exist $a_1,a_2$ with $$a_1+a_2> n+f(r)\log n$$ such that $a_1!a_2! \mid n!2^n3^n\cdots p_r^n$?

Is there some function $f(r)$ such that $f(r)\to \infty$ as $r\to\infty$, such that, for infinitely many $n$, there exist $a_1,a_2$ with $$a_1+a_2> n+f(r)\log n$$ such that $a_1!a_2! \mid n!2^n3^n\cdots p_r^n$?

It is ambiguous in [ErGr80] what the intended quantifiers are on the variables (they write 'is it true that we can find $a_1+a_2>n+f(r)\log n$...'). Comparing to previous problems such as [728] and [729] it seems most likely that they intended to ask the formulation in the problem statement.

The answer is yes: Barreto and Leeham have used ChatGPT to provide a proof of the stated problem (in fact essentially the same construction as their solution to [729]).