Ramanujan prime

In 1919, Ramanujan published a new proof of Bertrand's postulate which, as he notes, was first proved by Chebyshev. At the end of the two-page published paper, Ramanujan derived a generalized result, and that is: : \pi(x) - \pi\left( \frac x 2 \right) \ge 1,2,3,4,5,\ldots \text{ for all } x \ge 2, 11, 17, 29, 41, \ldots \text{ respectively}     where \pi(x) is the prime-counting function, equal to the number of primes less than or equal to x. The converse of this result is the definition of Ramanujan primes: :The nth Ramanujan prime is the least integer Rn for which \pi(x) - \pi(x/2) \ge n, for all x ≥ Rn. In other words: Ramanujan primes are the least integers Rn for which there are at least n primes between x and x/2 for all x ≥ Rn. The first five Ramanujan primes are thus 2, 11, 17, 29, and 41. Note that the integer Rn is necessarily a prime number: \pi(x) - \pi(x/2) and, hence, \pi(x) must increase by obtaining another prime at x = Rn. Since \pi(x) - \pi(x/2) can increase by at most 1, : \pi(R_n) - \pi\left( \frac{R_n} 2 \right) = n. ==Bounds and an asymptotic formula==

For all n \geq 1, the bounds :2n\ln2n hold. If n > 1, then also :p_{2n} where pn is the nth prime number. As n tends to infinity, Rn is asymptotic to the 2nth prime, i.e., :Rn ~ p2n (n → ∞). All these results were proved by Sondow (2009), except for the upper bound Rn 3n which was conjectured by him and proved by Laishram (2010). The bound was improved by Sondow, Nicholson, and Noe (2011) to :R_n \le \frac{41}{47} \ p_{3n} which is the optimal form of Rn ≤ c·p3n since it is an equality for n = 5. ==References==