Let $a,b,n$ be natural numbers such that $a!b! | n!$. I am looking for a (somehow best) upper bound of $a+b$ in terms of $n$ (for large values on $n$). For example it is clear to see that we must have $a+b \leq 2n$. But unfortunately I am looking for much smaller bound ! Any idea would be helpful.

Let $a,b,n$ be natural numbers such that $a!b! \mid n!$. I am looking for a (somehow best) upper bound of $a+b$ in terms of $n$ (for large values on $n$). For example it is clear to see that we must have $a+b \leq 2n$. But unfortunately I am looking for much smaller bound ! Any idea would be helpful.