I believe proving (or disproving) such a statement is beyond current technology. On the other hand, by a crude heuristics the conjecture must be right: $R_1,\dots,R_n$ are $n$ odd numbers up to $\sim 3n\ln(n)$ which are rather evenly distributed in size and in residue classes. In this range the density of primes among odd numbers is $\sim 2/\ln(n)$, so $R(n)$ should be $\sim 2n/\ln(n)$.
I believe proving (or disproving) such a statement is beyond current technology. On the other hand, by a crude heuristics the conjecture must be right: $R_1,\dots,R_n$ are $n$ odd numbers up to $\sim 3n\ln(n)$ which are rather evenly distributed in size and in residue classes. In this range the density of primes among odd numbers is $\sim 2/\ln(n)$, so $R(n)$ should be $\sim 2n/\ln(n)$.