I've stumbled across this playing around and summing primes at random during a boring lecture. Is this a known conjecture? Can it be proven?
My conjecture: There exists at least one non trivial solution such that $2p_n = p_a + p_b$ (the trivial being obviously $a=b=n$) for $n > 2$.
Tested by starting at the trivial $2p_n = p_n + p_n$ then incremented the right as I decremented the left until both were prime again or I've ran below $2$ with the left number. The second condition never occurred though, and I've tested for the first $1000$ primes by writing a simple program.
It fascinated me for the fact that this would mean that $p_b = 2p_n - p_a$ where $b > n > a$ so by knowing primes up to the $n$th you would have enough information to evaluate the $(n + 1)$-th?