Are there Infinity many natural numbers (that are not factorials) that divide all factorials (greater than that number) evenly?

Question

For example:

If n = 12

Then starting at 4! because 3! is less than 12

(4!) % 12 == 0

Therefore (4!) * 5 = (5!) % 12 == 0

etc

and so on until (n-1!) because every factorial after that would obviously be divisible by n.

This is not easy for my (extremely out of date)computer to solve so here is a short list of numbers I have found this to be true for:

1,2,3,6,8,12,24,30,40,60,120,144,180,240,360,720,840,1008,1260,1680,2520, 5040,5760,6720

*arguably 1 2 and 3 should not be included in this list

Answer 1

Suppose $(n-1)!<N\le n!$ and $n!=mN$. Then $$ \frac{m}{n} < 1 \le m $$ and so, apart from 1, these are the numbers of the form $n!/k$ where $k<n,$ given in A058298.

Answer 2

For $n=12$, you might have stopped with $4!$ right away, because of course $5!=5\cdot 4!$ etc.

First of all, there are infinitely many numbers $n$ such that $k!>n$ does not always imply $n\mid k!$. Just consider the case that $n$ is prime, which requires $k\ge n$.

Just as simply, there are infinitely many numbers $n$ for which $k!\ge n$ implies $n\mid k!$. In fact, this is certainly true if $n$ itself is a factorial.