Consider: $$q=\left\lfloor (2 p_n+p_{n+2}) (p_{n}+p_{n+1}+p_{n+2})\over p_{n}\right\rfloor,\;\text{isPrime}(q)$$
where $p_n$ denotes the $n$th prime.
Other than: $$n=6\quad\text{or}\quad n=8\quad\text{or}\quad n=9$$ $$p_6=17\quad\text{or}\quad p_8=23\quad\text{or}\quad p_9=29$$
Are there any $n$ such that $\text{isPrime(q)}=\text{true}$?
I have tested all $p_n < 10^7$, so if someone could explain why this is the case(if it is the case), or present an unconditional argument; It would be greatly appreciated.
Cheers!
Edit:
Observation:
Most $q,\;q>(p_9p_{10})$ is of form such that $q=3k$