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\vskip 1cm{\LARGE\bf 
Ramanujan Primes: \\
\vskip .1in
Bounds, Runs, Twins, and Gaps 
}
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\large
Jonathan~Sondow\\
209~West~97th~Street\\
New York,~NY~~10025\\
USA\\
\href{mailto:jsondow@alumni.princeton.edu}{\tt jsondow@alumni.princeton.edu} \\
\ \\
John~W.~Nicholson\\
P.~O.~Box~2423\\
Arlington,~TX~~76004\\
USA\\
\href{mailto:reddwarf2956@yahoo.com}{\tt reddwarf2956@yahoo.com} \\
\ \\
Tony~D.~Noe\\
14025~NW~Harvest~Lane\\
Portland,~OR~~97229\\
USA\\
\href{mailto:noe@sspectra.com}{\tt noe@sspectra.com} \\
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\begin{abstract}
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 $R_1,R_2,\dotsc,R_n$.\\
\end{abstract}


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