Let $n$ an integer sufficiently large.

I'm looking for a "big" set $S$ of prime number $p$ satisfying $10.n \geq p > \sqrt{18n} $ and $p| C_{9.n}^{k-n}C_{8n+k}^{8.n}$ for  all integer $k$ satifying $  n \leq  k \leq 10n$ and $k \geq p $ 

that means I'm looking a heavy condition  $(C)$ over the $p$ prime $10n \geq p > \sqrt{18n} $  giving me for all $p$ prime satisfying $(C)$ we have for all integers $k$ such that $\max(n,p) \leq  k \leq 10n$ one has

$$v_p(C_{9n}^{k-n}C_{8n+k}^{8n})=\Big[\frac{8n+k}{p}\Big]-\Big[\frac{k}{p}\Big]-\Big[\frac{8n}{p}\Big]+\Big[\frac{9n}{p}\Big]-\Big[\frac{k-n}{p}\Big]-\Big[\frac{10n-k}{p}\Big] \geq 1$$

where $[x]$ is the integer part of a real number $x$.


I think $(C) $ is of  type $p \geq \sqrt{18n}, \frac{a}{b}<\frac{n}{p}-[\frac{n}{p}]< \frac{c}{d} $for some integer $a,b,c,d $ that I coudn't find, which gives us the existence of an integer $h$ such that $p \in \Big]\displaystyle \frac{dn}{c+dh}, \displaystyle \frac{nb}{a+rh}\Big[$.

I have proved that for all $p \in(8n,9n),$ one has $v_p(C_{9n}^{k-n}C_{8n+k}^{8n})\ge 1$. There are many others prime number satisfying $v_p(C_{9n}^{k-n}C_{8n+k}^{8n})\ge 1$, maybe in $\big(\frac{8n}{2},\frac{9n}{2}\big), \big(\frac{8n}{3},\frac{9n}{3}\big)...$. 

I don't know what the $a,b,c,d$ are. Numerically, I have found many $p$ satisfying $v_p(C_{9n}^{k-n}C_{8n+k}^{8n})\ge 1$ and $10n \geq p > \sqrt{18n}$.

Any help is appreciated.