Sequence of Integers whose Factorial plus 1 is Prime

Sequence
The sequence of positive integers $n$ for which $n! + 1$ is prime begins:
 * $0, 1, 2, 3, 11, 27, 37, 41, 73, 77, 116, 154, 320, 340, 399, 427, 872, 1477, 6380, 26 \, 951, 110 \, 059, 150 \, 209 \ldots$

The sequence of corresponding primes begins:
 * $2, 2, 3, 7, 39 \, 916 \, 801, \ldots$

Also see

 * Sequence of Integers whose Factorial minus 1 is Prime