th Ramanujan prime is the smallest positive integer 
such that if 
, then the interval
![$ \left(\frac12x,x\right]$](img5.gif){kind=small}
contains at least primes. We sharpen Laishram's theorem that
by proving that the maximum of
is
. We give statistics on the length of the longest run of
Ramanujan primes among all primes 
, for 
. We prove that
if an upper twin prime is Ramanujan, then so is the lower; a table
gives the number of twin primes below 
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
.