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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
Software:
OEIS
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