

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
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OFFSET

1,1


COMMENTS

a(n) is the smallest prime on the right side of the Ramanujan Prime Corollary, 2*p_(in) > p_i, for i > k where k = pi(p_k) = pi(R_n) That is, p_k is the nth Ramanujan Prime, R_n and the kth 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_(in)] is greater than or equal to 1. By taking into account the size of the gaps between primes in [p_(in),p_k], one can see that the average prime gap is about log(p_k) using the following R_n / (2*n) ~ log(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_(kn) < (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_(kn) < (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), 181182.
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) 630635.
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 nth Ramanujan prime is A104272(n)= 97, the value of k = 25, so i is >= 26, in >= 16, the in 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.


PROG

(Perl) use ntheory ":all"; say next_prime(nth_ramanujan_prime($_)) for 1..100; # Dana Jacobsen, Dec 25 2015


CROSSREFS

Cf. A104272, A168421, A179196, A190874.
Cf. A202187, A202188, A234298.
Sequence in context: A260802 A045435 A038974 * A252090 A079419 A117300
Adjacent sequences: A168422 A168423 A168424 * A168426 A168427 A168428


KEYWORD

nonn


AUTHOR

John W. Nicholson, Nov 25 2009


STATUS

approved



