The first prime is $2$. The second prime is $3$. $3\cdot 2 = 6$. The product of the first three primes is $30$. The product of all the primes up to the fourth prime is $210$.
My question is this: Is this sequence described in words as the products of all primes up to the $n$th prime?
The first few members of this sequence are $2$, $6$, $30$, $210$, $2310$, $30030$, $510510\dots$
Described in words, this sequence is called the Primorials (prime + factorial) and is often written as $$\text{prime}(n)\#,$$ but note that there isn't anything wrong with your terminology: "products of all primes up to the $n$th prime."