Summary: The $ n$th Ramanujan prime is the smallest positive integer $ R_n$ such that if $ x \ge R_n$, then the interval $ \left(\frac12x,x\right]$ contains at least $ n$ primes. We sharpen Laishram's theorem that $ R_n < p_{3n}$ by proving that the maximum of $ R_n/p_{3n}$ is $ R_5/p_{15} = 41/47$. We give statistics on the length of the longest run of Ramanujan primes among all primes $ p<10^n$, for $ n\le9$. We prove that if an upper twin prime is Ramanujan, then so is the lower; a table gives the number of twin primes below $ 10^n$ of three types. Finally, we relate runs of Ramanujan primes to prime gaps. Along the way we state several conjectures and open problems. An appendix explains Noe's fast algorithm for computing $ R_1,R_2,\dotsc,R_n$.

11A41

prime gap, Ramanujan prime, twin prime