When is it true that $a^{n}How can I  find the smaller $n\in\mathbb{N}$, which makes the equation true: $$a^{n}<n!$$
For example:
If $a=2$ then $\longrightarrow 2^{n}<n!$ when $n\geq 4$
If $a=3$ then $\longrightarrow 3^{n}<n!$ when $n\geq 7$
If $a=4$ then $\longrightarrow 4^{n}<n!$ when $n\geq 9$
If $a=5$ then $\longrightarrow 5^{n}<n!$ when $n\geq 12$
$$a,n\in\mathbb{N}$$
$$a^{n}<n!$$when$$n\geq ?$$

I find a equation:

$$ a^{n}<n! \Longleftrightarrow n \geq  (2a)+\lfloor\frac{(a-1)}{2}\rfloor$$

