Consecutive Integers whose Product is Primorial

Theorem
The following primorials can be expressed as the product of consecutive integers:


 * $2, 6, 30, 210, 510 \, 510$

No others are known.

The corresponding indices of those primorials are:
 * $2, 3, 5, 7, 17$

The corresponding values of $n$ such that $p\# = \paren {n - 1} n$ are:
 * $2, 3, 6, 15, 715$