Sequence of Prime Primorial plus 1

Theorem
Let $p_n \#$ denote the $n$th primorial.

The sequence $\left\langle{n}\right\rangle$ such that $p_n \# + 1$ is prime begins:
 * $2, 3, 5, 7, 11, 31, 379, 1019, 1021, 2657, 3229, 4547, 4787, 11549, 13649, 18523, 23801, 24029, 42209, 145823, 366439, 392113, \ldots$