  The $n$th Ramanujan prime is the smallest positive integer $R_n$ such that if
$x \ge R_n$, then there are at least $n$ primes in the interval $(x/2,x]$. For
example, Bertrand's postulate is $R_1 = 2$. Ramanujan proved that $R_n$ exists
and gave the first five values as 2, 11, 17, 29, 41. In this note, we use
inequalities of Rosser and Schoenfeld to prove that $2n \log 2n < R_n < 4n \log
4n$ for all $n$, and we use the Prime Number Theorem to show that $R_n$ is
asymptotic to the $2n$th prime. We also estimate the length of the longest
string of consecutive Ramanujan primes among the first $n$ primes, explain why
there are more twin Ramanujan primes than expected, and make three conjectures
(the first has since been proved by S. Laishram).
