Is it known when the summation is equal to factorial ?
$$ n! = \sum_{i=1}^{m}i = \frac{m(m+1)}{2} $$
where $n,m \in \mathbb{N}$
$$ 3! = \sum_{i=1}^{3}i = 6 $$
$$ 5! = \sum_{i=1}^{15}i = 120 $$
And also when product of primes is equal to summation:
$$ \prod_{i=1}^{n}P_i = \sum_{i=1}^{m}i $$
where $P_n$ is the $n$th prime and $m \in \mathbb{N}$
$$ \prod_{i=1}^{2}P_i = 2\times3 = \sum_{i=1}^{2}i = 6 $$
$$ \prod_{i=1}^{4}P_i = 2\times3\times5\times7 = \sum_{i=1}^{20}i = 210 $$
I was not able to find other pairs of factorial and primorial which are equals to summation.
I would like to know how to prove that other pairs either exist or doesn't exist.