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A168421
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Small Associated Ramanujan Prime, p_(i-n)
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6
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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
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OFFSET
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1,1
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COMMENTS
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a(n) is the left side of the Ramanujan Prime Corollary
2*p_(i-n) > p_i
for i > k where k = pi(p_k) = pi(R_n) That is, p_k is the n'th Ramanujan Prime, R_n and the k'th prime.
Smallest prime number, a(n), such that if x >= a(n), then there are at least n primes between x and 2x exclusively.
This is very useful in showing the number of primes in the range [p_k, 2*p_(i-n)] is greater than or equal to 1. By taking into account the size of the gaps between primes in [p_(i-n),p_k], one can see that the average prime gap is about ln(p_k) using the following R_n / (2*n) ~ ln(R_n).
Proof of Corollary: See Wikipedia link
The number of primes until the next Ramanujan prime, R_(n+1), can be found in A190874.
Not the same as A124136.
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LINKS
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T. D. Noe, Table of n, a(n) for n = 1..10000
J. Sondow, Ramanujan primes and Bertrand's postulate, Amer. Math. Monthly 116 (2009) 630-635.
J. Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, J. Integer Seq. 14 (2011) Article 11.6.2
Wikipedia, Ramanujan Prime
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FORMULA
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a(n) = prime(primepi(A104272(n)) + 1 - n)
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EXAMPLE
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For n=10, the n'th Ramanujan prime is A104272(n)= 97, the value of k = 25, so i is >= 26, i-n >= 16, the i-n prime is 53, and 2*53 = 106. This leaves the range [97, 106] for the 26th prime which is 101. In this example, 53 is the small associated Ramanujan prime.
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CROSSREFS
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Cf. A104272, A168425, A179196, A190874.
Sequence in context: A063205 A090613 A063097 * A038942 A175283 A124136
Adjacent sequences: A168418 A168419 A168420 * A168422 A168423 A168424
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KEYWORD
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nonn
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AUTHOR
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John W. Nicholson, Nov 25 2009
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EXTENSIONS
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Extended by T. D. Noe, Nov 22 2010
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STATUS
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approved
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