A168425 Large Associated Ramanujan Prime, p_i.

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

Offset: 1

Author

John W. Nicholson, Nov 25 2009

Keywords

• nonn

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 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_(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

Examples

			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.
		

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.
• Vladimir Shevelev, Ramanujan and Labos primes, their generalizations and classifications of primes, arXiv:0909.0715 [math.NT], 2009-2011.
• Jonathan Sondow, Ramanujan primes and Bertrand's postulate, Amer. Math. Monthly 116 (2009) 630-635.
• Jonathan Sondow, John W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, J. Integer Seq. 14 (2011) Article 11.6.2.
• Jonathan Sondow, Ramanujan Prime in MathWorld
• Anitha Srinivasan, An upper bound for Ramanujan primes, Integers, 19 (2014), #A19.
• Wikipedia, Ramanujan Prime

Crossrefs

Cf. A104272, A168421, A179196, A190874.

Cf. A202187, A202188, A234298.

Programs

Mathematica

nn = 100; R = Table[0, {nn}]; s = 0;
Do[If[PrimeQ[k], s++]; If[PrimeQ[k/2], s--];
If[s < nn, R[[s+1]] = k], {k, Prime[3 nn]}
];
RamanujanPrimes = R + 1;
Prime[PrimePi[#]+1]& /@ RamanujanPrimes (* Jean-François Alcover, Nov 03 2018, after T. D. Noe in A104272 *)

PARI

genit(n=100)={my(L=vector(n),s=0,k=1,z);for(k=1,prime(3*n)-1,if(ispseudoprime(k),s++);if(k%2==0&&ispseudoprime(k/2),s--);if(snextprime(x+1),L);v} \\ Bill McEachen, Jun 24 2023 (incorporates code from A104272)

Perl

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

Formula

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

a(n) = A151800(A104272(n)). - Michel Marcus, Jun 27 2023