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%I #78 Sep 03 2023 10:39:09
%S 3,13,19,31,43,53,61,71,73,101,103,109,131,151,157,173,181,191,229,
%T 233,239,241,251,269,271,283,311,313,349,353,373,379,409,419,421,433,
%U 439,443,463,491,499,509,571,577,593,599,601,607,613,643,647,653,659,661
%N Large Associated Ramanujan Prime, p_i.
%C 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.
%C a(n) = nextprime(R_n) = nextprime(p_k), where nextprime(x) is the next prime larger than x.
%C 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).
%C Proof of Corollary: See Wikipedia link.
%C The number of primes until the next Ramanujan prime, R_(n+1), can be found in A190874.
%C 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
%C 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
%H T. D. Noe, <a href="/A168425/b168425.txt">Table of n, a(n) for n = 1..1000</a>
%H S. Ramanujan, <a href="http://ramanujan.sirinudi.org/Volumes/published/ram24.html">A proof of Bertrand's postulate</a>, J. Indian Math. Soc., 11 (1919), 181-182.
%H Vladimir Shevelev, <a href="http://arxiv.org/abs/0909.0715">Ramanujan and Labos primes, their generalizations and classifications of primes</a>, arXiv:0909.0715 [math.NT], 2009-2011.
%H Jonathan Sondow, <a href="http://arxiv.org/abs/0907.5232">Ramanujan primes and Bertrand's postulate</a>, Amer. Math. Monthly 116 (2009) 630-635.
%H Jonathan Sondow, John W. Nicholson, and T. D. Noe, <a href="http://arxiv.org/abs/1105.2249"> Ramanujan Primes: Bounds, Runs, Twins, and Gaps</a>, J. Integer Seq. 14 (2011) Article 11.6.2.
%H Jonathan Sondow, <a href="http://mathworld.wolfram.com/RamanujanPrime.html">Ramanujan Prime in MathWorld</a>
%H Anitha Srinivasan, <a href="http://www.emis.de/journals/INTEGERS/papers/o19/o19.Abstract.html">An upper bound for Ramanujan primes</a>, Integers, 19 (2014), #A19.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Ramanujan_prime">Ramanujan Prime</a>
%F a(n) = prime(primepi(A104272(n)) + 1).
%F a(n) = A151800(A104272(n)). - _Michel Marcus_, Jun 27 2023
%e 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.
%t nn = 100; R = Table[0, {nn}]; s = 0;
%t Do[If[PrimeQ[k], s++]; If[PrimeQ[k/2], s--];
%t If[s < nn, R[[s+1]] = k], {k, Prime[3 nn]}
%t ];
%t RamanujanPrimes = R + 1;
%t Prime[PrimePi[#]+1]& /@ RamanujanPrimes (* _Jean-François Alcover_, Nov 03 2018, after _T. D. Noe_ in A104272 *)
%o (Perl) use ntheory ":all"; say next_prime(nth_ramanujan_prime($_)) for 1..100; # _Dana Jacobsen_, Dec 25 2015
%o (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(s<n,L[s+1]=k+1));v=apply(x->nextprime(x+1),L);v} \\ _Bill McEachen_, Jun 24 2023 (incorporates code from A104272)
%Y Cf. A104272, A168421, A179196, A190874.
%Y Cf. A202187, A202188, A234298.
%K nonn
%O 1,1
%A _John W. Nicholson_, Nov 25 2009
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