Since $2+3=5$ I know a simle example of a triple of coprime integers $(a,b,c)$, such that $$a+b=c,$$ being each summand a product of distinct primes $a=2$, $b=3$ and $c=5$, and their product is a primorial $$abc=2\cdot3\cdot 5.$$ It is the primorial or order $3$, that is the positive integer $30$.
I believe that the following problem should be in the literaute but I've curiosity about its solution
Question. What about the triples of (coprime) integers $(a,b,c)$ being solutions of the equation $$\underbrace{\text{product of distinct primes}}_{a}+\underbrace{\text{product of distinct primes}}_{b}=\underbrace{\text{product of distinct primes}}_{c}$$ that satisfy the condition $$abc=\text{a primorial}?$$ What I am asking is if it is known that there exists a $K$ such that $\forall k\geq K$ there are no solution of our previous problem, o well in other case there are infinitely many solutions. Many thanks.
My belief is that should be few solutions. If this problem was in the literature refer the literature as an answer, and I try find the article.
Edit: I didn't know such sequence ,and I didn't calculate terms of such. Many thanks @RobertIsrael for your reference to OEIS, I add here thus the name of the author of the sequence Naohiro Nomoto (Sep 10 2000), and more terms were added by Carlos Rivera.
For order $3$: $2+3=5$.
For order $4$: $3 + 7 = 10$.
For order $5$: $2 + 33 = 35$.
For order $6$: $13 + 42 = 55$.
For order $7$: $11 + 210 = 221$.
For order $8$: $57 + 385 = 442$.
For order $9$: $119 + 1311 = 1430$.
For order $10$: $1495 + 1463 = 2958$.
For order $11$: $374 + 22971 = 23345$.
For order $13$: $1235 + 495726 = 496961$.
EDIT: See OEIS sequence A057035.