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Ramanujan primes: bounds, runs, twins, and gaps. (English) Zbl 1229.11014
The $$n$$th Ramanujan prime is the smallest positive integer $$R_n$$ such that if $$x \geq R_n$$, then the interval $$(\frac{1}{2}x,x]$$ contains at least $$n$$ primes. The authors prove that the maximum of $$R_n/p_{3n}$$ is $$R_5/p_{15} = 41/47$$. They present statistics on the length of the longest run of Ramanujan primes among all primes $$p<10^n$$, for $$n\leq9$$. If an upper twin prime is Ramanujan, then so is the lower. Runs of Ramanujan primes are related to prime gaps. An appendix explains Noe’s fast algorithm for computing $$R_1,R_2,\dots ,R_n$$.

##### MSC:
 11A41 Primes
##### Keywords:
Ramanujan prime; twin prime; prime gap
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