the sequence $n!+2,...,n!+n$ is made up of only composite numbers I have found the claim that given that $n\geq2$, we have that the sequence of $n-1$ numbers $n!+2,n!+3,...,n!$ is made up of only composite numbers. Is there a proof of this? I found this pretty fascinating but I am not sure how to go around it. It seems to hold for the first few examples
$n=2$:  $S=\{4\}$
$n=3$: $S=\{8,9\}$
$n=4$: $S=\{26,27,28\}$

For a given $n$, the $i$-th term is:
$$
i+n!=i(1+1\times 2 \times \cdots \times (i-1) \cdots (i+1)\times \cdots n)
$$
Thus the $i$-th term is the product of $i$ and $(1+n!/i)$, thus it is a composite number.
