Erdős Problem #401

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]).