Sequence of Integers whose Factorial minus 1 is Prime

Sequence
The sequence of positive integers $n$ for which $n! - 1$ is prime begins:
 * $3, 4, 6, 7, 12, 14, 30, 32, 33, 38, 94, 166, 324, 379, 469, 546, 974, 1963, 3507, 3610, 6917, 21 \, 480, 34 \, 790, 94 \, 550, 103 \, 040, 147 \, 855, 208 \, 003 \ldots$

The sequence of corresponding primes begins:
 * $5, 23, 719, 5039, 479 \, 001 \, 599, 87 \, 178 \, 291 \, 199, \ldots$

Also see

 * Sequence of Integers whose Factorial plus 1 is Prime