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# Ramanujan prime

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In mathematics, a **Ramanujan prime** is a prime number that satisfies a result proven by Srinivasa Ramanujan. It relates to the prime counting function.

## Origins and definition

In 1919, Ramanujan published a new proof of Bertrand's postulate (which had already been proven by Pafnuty Chebyshev).

Ramanujan's result at the end of the paper was:

- [math]\pi(x) - \pi(x/2)[/math] ≥ 1, 2, 3, 4, 5, ... for all
*x*≥ 2, 11, 17, 29, 41, ... (sequence A104272 in OEIS)

where [math]\pi[/math]*(x)* is the prime counting function. The prime counting function is the number of primes less than or equal to *x*.

The numbers 2, 11, 17, 29, 41 are first few Ramanujan primes. In other words:

Ramanujan primes are the integers *R _{n}* that are the

**smallest**to satisfy the condition

- [math]\pi(x) - \pi(x/2)[/math] ≥
*n*, for all*x*≥*R*_{n}