Ramanujan primes: bounds, runs, twins, and gaps
Journal of integer sequences, Tome 14 (2011) no. 6
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$.
@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},
year = {2011},
volume = {14},
number = {6},
zbl = {1229.11014},
language = {en},
url = {http://geodesic.mathdoc.fr/item/JIS_2011__14_6_a7/}
}
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/