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A168425 Large Associated Ramanujan Prime, p_i 10
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

a(n) is the smallest prime on 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.

a(n) = nextprime(R_n) = nextprime(p_k), where nextprime(x) is the next prime larger than x.

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.

Srinivasan's Lemma (2014): p_(k-n) < (p_k)/2 if R_n = p_k and n > 1. Proof: By the minimality of R_n, the interval ((p_k)/2,p_k] contains exactly n primes, so p_(k-n) < (p_k)/2. - Jonathan Sondow, May 10 2014

In spite of the name Large Associated Ramanujan Prime, a(n) is not a Ramanujan prime for many values of n. - Jonathan Sondow, May 10 2014

LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000

S. Ramanujan, A proof of Bertrand's postulate, J. Indian Math. Soc., 11 (1919), 181-182.

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

Anitha Srinivasan, An upper bound for Ramanujan primes, Integers, 19 (2014), #A19

Wikipedia , Ramanujan Prime

FORMULA

a(n) = prime(primepi(A104272(n)) + 1)

EXAMPLE

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.

CROSSREFS

Cf. A104272, A168421, A179196, A190874.

Cf. A202187, A202188, A234298.

Sequence in context: A071600 A045435 A038974 * A079419 A117300 A023215

Adjacent sequences:  A168422 A168423 A168424 * A168426 A168427 A168428

KEYWORD

nonn

AUTHOR

John W. Nicholson, Nov 25 2009

STATUS

approved

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Last modified September 19 18:00 EDT 2014. Contains 246977 sequences.