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where ''p''<sub>''n''/(1 − ''c'')</sub> is the <math>\lfloor</math>''n''/(1 − ''c'')<math>\rfloor </math>th prime and <math>\lfloor \cdot\rfloor</math> is the [[Floor and ceiling functions|floor]] function.
== Ramanujan prime corollary ==
:<math>2p_{i-n} > p_i \text{ for } i>k \text{ where } k=\pi(p_k)=\pi(R_n)\, ,</math>
i.e. ''p''<sub>''k''</sub> is the ''k''th prime and the ''n''th Ramanujan prime.
This is very useful in showing the number of primes in the range [''p''<sub>''k''</sub>, 2''p''<sub>''i''−''n''</sub>] is greater than or equal to 1. By taking into account the size of the gaps between primes in [''p''<sub>''i''−''n''</sub>,''p''<sub>''k''</sub>], one can see that the average prime gap is about ln(''p''<sub>''k''</sub>) using the following ''R''<sub>''n''</sub>/(2''n'') ~ ln(''R''<sub>''n''</sub>). Note that all [[Prime_gap#Numerical_results|maximal prime pairs]] are bounded by, for a particular n and i, the prime ''p''<sub>''k''</sub> and twice the prime ''p''<sub>''i''-''n''</sub>. This means the gap between maximal prime pair can not more than double the prior maximal prime pair.
Proof of Corollary:
: If ''p''<sub>''i''</sub> > ''R''<sub>''n''</sub>, then ''p''<sub>''i''</sub> is odd and ''p''<sub>''i''</sub> − 1 ≥ ''R''<sub>''n''</sub>, and hence {{pi}}(''p''<sub>''i''</sub> − 1) − {{pi}}(''p''<sub>''i''</sub>/2) = {{pi}}(''p''<sub>''i''</sub> − 1) − {{pi}}((''p''<sub>''i''</sub> − 1)/2) ≥ ''n''.
: Thus ''p''<sub>''i''</sub> − 1 ≥ ''p''<sub>''i''−1</sub> > ''p''<sub>''i''−2</sub> > ''p''<sub>''i''−3</sub> > ... > ''p''<sub>''i''−''n''</sub> > ''p''<sub>''i''</sub>/2, and so 2''p''<sub>''i''−''n''</sub> > ''p''<sub>''i''</sub>.
An example of this corollary:
With ''n'' = 1000, ''R''<sub>''n''</sub> = ''p''<sub>''k''</sub> = 19403, and ''k'' = 2197, therefore ''i'' ≥ 2198 and ''i''−''n'' ≥ 1198.
The smallest ''i'' − ''n'' prime is ''p''<sub>''i''−''n''</sub> = 9719, therefore 2''p''<sub>''i''−''n''</sub> = 2 × 9719 = 19438. The 2198th prime, ''p''<sub>''i''</sub>, is between ''p''<sub>''k''</sub> = 19403 and 2''p''<sub>''i''−''n''</sub> = 19438 and is 19417.
The left side of the Ramanujan Prime Corollary is the {{OEIS2C|id=A168421}}; the smallest prime on the right side is {{OEIS2C|id=A168425}}. The sequence {{OEIS2C|id=A165959}} is the range of the smallest prime greater than p<sub>k</sub>. The values of <math>\pi(R_n)\,</math> are in the {{OEIS2C|id=A179196}}.
The Ramanujan Prime Corollary is due to John Nicholson.
Srinivasan's Lemma <ref>{{Citation |first=Anitha |last=Srinivasan |title=An upper bound for Ramanujan primes |journal=Integers |volume=14 |year=2014 |url=http://www.emis.ams.org/journals/INTEGERS/papers/o19/o19.pdf }}</ref> states that ''p''<sub>''k''−''n''</sub> < ''p''<sub>''k''</sub>/2 if ''R''<sub>''n''</sub> = ''p''<sub>''k''</sub> and ''n'' > 1. Proof: By the minimality of ''R''<sub>''n''</sub>, the interval (''p''<sub>''k''</sub>/2,''p''<sub>''k''</sub>] contains exactly ''n'' primes and hence ''p''<sub>''k''−''n''</sub> < ''p''<sub>''k''</sub>/2.
==References==
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