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

Ramanujan Primes: Bounds, Runs, Twins, and Gaps

Jonathan Sondow
209 West 97th Street
New York, NY 10025

John W. Nicholson
P. O. Box 2423
Arlington, TX 76004

Tony D. Noe
14025 NW Harvest Lane
Portland, OR 97229


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$.

Full version:  pdf,    dvi,    ps,    latex    

(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.

Return to Journal of Integer Sequences home page