Pillai prime

If for a given prime $p$ we can find an integer $n>0$ such that $n!\equiv -1modp$ but $p\not\equiv 1modn$ then $p$ is a called a Pillai prime. These are listed in A063980 of Sloane’s OEIS. Sarinya Intaraprasert proved that there are infinitely many Pillai primes. The first few are 23, 29, 59, 61, 67, 71, 79, 83, 109, 137, 139, 149, 193, …