Geodesic
Ramanujan primes: bounds, runs, twins, and gaps
Journal of integer sequences, Tome 14 (2011) no. 6.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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 $ p10^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$.
Classification : 11A41
Keywords: prime gap, Ramanujan prime, twin prime 
@article{JIS_2011__14_6_a7,
     author = {Sondow, Jonathan and Nicholson, John W. and Noe, Tony D.},
     title = {Ramanujan primes: bounds, runs, twins, and gaps},
     journal = {Journal of integer sequences},
     publisher = {mathdoc},
     volume = {14},
     number = {6},
     year = {2011},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2011__14_6_a7/}
}
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Sondow, Jonathan; Nicholson, John W.; Noe, Tony D. Ramanujan primes: bounds, runs, twins, and gaps. Journal of integer sequences, Tome 14 (2011) no. 6. http://geodesic.mathdoc.fr/item/JIS_2011__14_6_a7/