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A168425
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Large Associated Ramanujan Prime, p_i
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9
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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, 439, 443, 463, 491, 499, 509, 571, 577, 593, 599, 601, 607, 613, 643, 647, 653, 659, 661
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OFFSET
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1,1
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COMMENTS
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a(n) is the right 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.
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.
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LINKS
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T. D. Noe, Table of n, a(n) for n = 1..1000
V. Shevelev, Ramanujan and Labos primes, their generalizations and classifications of primes
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
J. Sondow, Ramanujan Prime in MathWorld
Wikipedia , Ramanujan Prime
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FORMULA
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a(n) = prime(primepi(A104272(n)) + 1)
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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, 101 is the large associated Ramanujan prime.
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CROSSREFS
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Cf. A104272, A168421, A179196, A190874.
Sequence in context: A071600 A045435 A038974 * A079419 A117300 A023215
Adjacent sequences: A168422 A168423 A168424 * A168426 A168427 A168428
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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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STATUS
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approved
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