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 $a_1!a_2! \mid n!2^n3^n\cdots p_r^n$?

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