Ramanujan primes

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A Ramanujan prime is a prime number that satisfies a result proved by Srinivasa Ramanujan relating to the prime counting function. 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.

The $\scriptstyle n\,$th Ramanujan prime is the least positive integer $\scriptstyle R_{n}\,$ for which

$\pi (x)-\pi ({\tfrac {x}{2}})\geq n,\,\forall x\geq R_{n},\quad n\geq 1,\,$

where $\scriptstyle \pi (x)\,$ is the prime counting function (number of primes less than or equal to $\scriptstyle x\,$).

A104272 Ramanujan primes R_n: a(n) is the smallest number such that if x >= a(n), then pi(x) - pi(x/2) >= n, where pi(x) is the number of primes <= x.

{2, 11, 17, 29, 41, 47, 59, 67, 71, 97, 101, 107, 127, 149, 151, 167, 179, 181, 227, 229, 233, 239, 241, 263, 269, 281, 307, 311, 347, 349, 367, 373, 401, 409, 419, 431, 433, ...}

Note that the integer R_n is necessarily a prime number: $\pi (x)-\pi (x/2)$ and, hence, $\pi (x)$ must increase by obtaining another prime at x = R_n. Since $\pi (x)-\pi (x/2)$ can increase by at most 1,

$\pi ($R_n$)-\pi ($R_n$/2)=n$.

A174635 Prime numbers that are not Ramanujan primes.

{3, 5, 7, 13, 19, 23, 31, 37, 43, 53, 61, 73, 79, 83, 89, 103, 109, 113, 131, 137, 139, 157, 163, 173, 191, 193, 197, 199, 211, 223, 251, 257, 271, 277, 283, 293, 313, 317, ...}

A193507 Ramanujan primes of the second kind: a(n) is the smallest prime such that if prime x >= a(n), then pi(x) - pi(x/2) >= n, where pi(x) is the number of primes <= x.

{2, 3, 13, 19, 31, 43, 53, 61, 71, 73, 101, 103, 109, 131, 151, 157, 173, 181, 191, 229, 233, 239, 241, 251, 269, 271, 283, 311, 313, 349, 353, 373, 379, 409, 419, 421, 433, ...}

A214934 Numbers R(k) such that R(k) >= 2k log R(k), where R(k) = A104272(k) is the k-th Ramanujan prime.

{2, 11, 17, 29, 41, 47, 59, 97, 127, 149, 151, 167, 179, 227, 229, 233, 347, 367, 401, 409, 569, 571, 587, 593, 937, 1423, 1427, 2237, 2617, 2657, 2659, 3251}

This is a complete list of Ramanujan prime numbers in which $R_{k}>=2klog(R_{k})$. For all $k>201$, $R_{k}<2klog(R_{k})$

Ramanujan Prime Corollary

The Ramanujan Prime Corollary, $2*p_{i-n}>p_{i}$ for $i>k$ where $k=\pi (p_{k})=\pi (R_{n})$, stems from the generalization of Bertrand's postulate. The power of this corollary comes from using a list of primes from $p_{i-n}$ to $R_{n}$ and knowing the value of $n$. Because $R_{n}<p_{i}$ and because of the index of $p_{i-n}$ the range to the mth prime after $R_{n}$ can be found with $2*p_{i-n+m}>p_{i+m}>p_{i+m-1}>...>p_{i}>R_{n}$.

The sequence A165959 is the difference $2*p_{i-n}-R_{n}$. If A165959 has an infinite number of terms in which a(n) = 3, then the twin prime conjecture can be proved.

A168421 Small Associated Ramanujan Prime, $p_{(i-n)}$, is the left side of the Ramanujan Prime Corollary

{2, 7, 11, 17, 23, 29, 31, 37, 37, 53, 53, 59, 67, 79, 79, 89, 97, 97, 127, 127, 127, 127, 127, 137, 137, 149, 157, 157, 179, 179, 191, 191, 211, 211, 211, 223, 223, 223, 233, 251, 251, 257, 293, 293, 307, 307, 307, 307, 307, 331, 331, 331, ...}

A168425 Large Associated Ramanujan Prime, p_i, is the right side of the Ramanujan Prime Corollary

{3, 13, 19, 31, 43, 53, 61, 71, 73, 101, 103, 109, 131, 151, 157, 173, 181, 191, 229, 233, 239, 241, 251, 269, 271, 283, 311, 313, 349, 353, 373, 379, 409, 419, 421, 433, ...}

A165959 Size of the range of the Ramanujan Prime Corollary, 2*A168421(n) - A104272(n)

{2, 3, 5, 5, 5, 11, 3, 7, 3, 9, 5, 11, 7, 9, 7, 11, 15, 13, 27, 25, 21, 15, 13, 11, 5, 17, 7, 3, 11, 9, 15, 9, 21, 13, 3, ...}