For the $n$-th prime number $p_n$ the primorial $p_n\#$ is defined as the product of the first $n$ primes.
What is the complexity of testing if a given number $n$ is a primorial?
Is it related in some way to the FACTORING problem?
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
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.