Does there exist an infinite number of values of a positive integers N such that N! is divisible by N^{2} + 1?

Give reasons for your answer.

Clearly N^2+1 must be composite. Furthermore, all the prime factors of N^2+1 are at most N. This is a necessary but not sufficient requirement (N=7 has N^2+1=50=2*5*5 but 50 does not divide 7!=5040).

After manually trying N up to 40, I found 18, 21, and 38 work. This was enough to find OEIS A120416, which has the first 36 such numbers.

I don't see any pattern in the sequence but values of N seem to be plentiful with the largest jump being 17, going from 21 to 38.