Complexity of testing if a number is a primorial
This is simpler than factoring.  You can simply multiply all prime numbers up to the point where you (a) reach $p_n$ or (b) get a product which is larger than $p_n$.  In the first case, you answer yes otherwise you answer no.

The reason it is simpler is that the prime factorization is trivial, whereas in the FACTORING problem, you don't know how many of each prime number you need.