Journal of Integer Sequences, Vol. 14 (2011), Article 11.6.2 |

209 West 97th Street

New York, NY 10025

USA

John W. Nicholson

P. O. Box 2423

Arlington, TX 76004

USA

Tony D. Noe

14025 NW Harvest Lane

Portland, OR 97229

USA

**Abstract:**

The th Ramanujan prime is the smallest positive integer
such that if , then the interval
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
.

(Concerned with sequences A007508 A065421 A104272 A173081 A174602 A174641 A177804 A178127 A178128 A179196 A181678 A189993 A189994.)

Received December 14 2010;
revised version received May 11 2011.
Published in *Journal of Integer Sequences*, May 17 2011.

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