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Ramanujan primes
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.[1] At the end of the two-page published paper, Ramanujan derived a generalized result.
The th Ramanujan prime is the least positive integer for which
where is the prime counting function (number of primes less than or equal to ).
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 Rn is necessarily a prime number: and, hence, must increase by obtaining another prime at x = Rn. Since can increase by at most 1,
- RnRn.
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 . For all ,
Ramanujan Prime Corollary
The Ramanujan Prime Corollary, for where , stems from the generalization of Bertrand's postulate. The power of this corollary comes from using a list of primes from to and knowing the value of . Because and because of the index of the range to the mth prime after can be found with .
The sequence A165959 is the difference . 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, , 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, ...}
Notes
- ↑ Ramanujan, S. (1919), “A proof of Bertrand's postulate”, Journal of the Indian Mathematical Society 11: 181–182.
- ↑ Sondow, Jonathan, Ramanujan Prime, from MathWorld—A Wolfram Web Resource, created by Eric W. Weisstein.