there are at least n primes between x and x/2 for all x >= R_n.

The first five Ramanujan primes are thus 2, 11, 17, 29, and 41.

Note that the integer R_n is necessarily a prime number: and, hence, must increase by obtaining another prime at x = R_n. Since can increase by at most 1,

Bounds and an asymptotic formula

For all , the bounds

hold. If , then also

where p_n is the nth prime number.

As n tends to

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