

A168421


Small Associated Ramanujan Prime, p_(in).


12



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

1,1


COMMENTS

a(n) is the smallest prime p_(k+1n) on the left side of the Ramanujan Prime Corollary, 2*p_(in) > p_i for i > k, where the nth Ramanujan Prime R_n is the kth prime p_k. [Comment clarified and shortened by Jonathan Sondow, Dec 20 2013]
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_(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.
Except for A000101(1)=3 and A000101(2)=5, A000101(k) = a(n). Because of the large size of a gap, there are many repeats of the prime number in this sequence.  John W. Nicholson, Dec 10 2013
For some n and k, we see that a(n) = A104272(k) as to form a chain of primes similar to a Cunningham chain. For example (and the first example), a(2) = 7, links A104272(2) = 11 = a(3), links A104272(3) = 17 = a(4), links A104272(4) = 29 = a(6), links A104272(6) = 47. Note that the links do not have to be of a form like q = 2*p+1 or q = 2*p1.  John W. Nicholson, Dec 14 2013
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 Small Associated Ramanujan Prime, a(n) is not a Ramanujan prime for many values of n.  Jonathan Sondow, May 10 2014
All maximal prime pairs in A002386 and A000101 are bounded by, for a particular n and i, the prime A104272(n) and twice a prime in A000040() following a(n). This means the gap between maximal prime pair cannot be more than twice the prior maximal prime gap.  John W. Nicholson, Feb 07 2019


LINKS



FORMULA

a(n) = prime(primepi(A104272(n)) + 1  n).
a(n) = nextprime(A084139(n+1)), where nextprime(x) is the next prime > x. Note: some A084139(n) may be prime, therefore nextprime(x) not equal to x.  John W. Nicholson, Oct 11 2013


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, 53 is the small associated Ramanujan prime.


MATHEMATICA

nn = 100;
t = Table[0, {nn}];
Do[m = PrimePi[2n]  PrimePi[n]; If[0 < m <= nn, t[[m]] = n], {n, 15 nn}];


PROG

(Perl) use ntheory ":all"; say next_prime((nth_ramanujan_prime($_)+1) >> 1) for 1..100; # Dana Jacobsen, Mar 02 2016


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



