Any relation between primorial numbers and oblong (n(n+1)) numbers?

Question

Just noticed that some primorial numbers are oblong:

$\prod\limits_{i=1}^{3}p_i = 5 \cdot 6$

$\prod\limits_{i=1}^{4}p_i = 14 \cdot 15$

$\prod\limits_{i=1}^{7}p_i = 714 \cdot 715$

Does anyone know if there are infinite cases of numbers that are at the same time oblong and primorial?

Answer 1

This PARI/GP program

(22:05) gp > n=1;s=1;while(n<10^6,n=nextprime(n+1);s=s*n;if(issquare(4*s+1)==1,print(n," ",s)))
2 2
3 6
5 30
7 210
17 510510
(22:11) gp >

shows that upt $10^6$# , there is no larger primorial than $510510$, which has the form $n(n+1)$. This leads to the strong conjecture that there are no more. To prove it, will be very difficult, I guess.