Are there infinitely many composiphiles?

Question

I came up with this today: we say a positive integer $k$ is a composiphile if there exists no positive integer $n \leq k$ such that $$\frac{k!}{n} + 1 \text{ is prime.}$$ My question: are there infinitely many composiphiles?

My comments: I have checked up to $k=50$, and have found $18$ to be the only composiphile no greater than $50$. I have tried a proof by contradiction to prove their infinitude, although it got very messy very quickly, and don't see how I might continue.

Update: just my luck, $51$ is also a composiphile, and so is $53$.

Answer 1

Using this self-defined function

comp(k)={gef=1;for(n=1,k,if(gef==1,if(ispseudoprime(k!/n+1)==1,gef=0)));gef==1}

we get the following start of the output

? for(s=1,500,if(comp(s)==1,print1(s," ")))
18 51 53 84 95 100 104 106 143 178 180 181 188 202 203 211 214 217 222 227 232 237 239 244 250 258 272 281 284 300 303 313 315 317 323 326 342 358 364 370 374 386 387 396 401 409 413 422 423 429

I aborted the calculation, hence the list upto 500 need not be complete. But upto 429, exactly the displayed numbers are composiphiles.

Heuristically, we can expect infinite many composiphiles. If $k$ is large, we have $k$ chances to find a prime with magnitude approximately $k!$. It is not a great surprise, if we do not find a prime, hence my guess. Of course, this is not a proof, but I have great doubts anyway that this can be actually proven (or, even more suprisingly , disproven).