Let $n,m$ be two positive integers. By $r\_n$ we denote the largest prime not exceeding $n$. If $r\_n\leq m\leq n$ and $q$ is the largest prime factor of $n!/m!$ such that $q\geq 17$ and $q\geq n-m+3$, then I would like to find some examples of integers $n$ and $m$ such that $q^4$ divides $n!$. 



Finding a suitable number
Given a positive integer $n$, let $P(n)$ denote the largest prime factor of $n$. What you are looking for are integers $n$ such that $q := \max\{P(n-15),P(n-14),\dots,P(n)\} < Cn$ for some constant $C \in ]0,1[$ depending on whether you require $q^2$ to divide $n!$ (as in the first version of your question) or $q^4$ like now or some higher power. However by


John G. Kemeny: [Largest prime factor](http://www.sciencedirect.com/science/article/pii/0022404993900939), *Journal of Pure and Applied Algebra* 89(1993) Issues 1-2, 181-186


the average value of $P(n)/n$ is asymptotically $\zeta(2)/\log n$, which tends to $0$ when $n$ tends to $\infty$. Therefore, regardless of whether you require $q^2$, $q^4$, $q^{10}$ or some higher power of $q$ to divide $n!$, the natural density of the set of the $n$ which fulfil your condition equals $1$.


A list of examples $(n,q,s)$ where $n \leq 10000$ is as follows:



```
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```

