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 A104272 Ramanujan primes R_n: a(n) is the smallest number such that if x >= a(n), then pi(x) - pi(x/2) >= n, where pi(x) is the number of primes <= x. 118

%I

%S 2,11,17,29,41,47,59,67,71,97,101,107,127,149,151,167,179,181,227,229,

%T 233,239,241,263,269,281,307,311,347,349,367,373,401,409,419,431,433,

%U 439,461,487,491,503,569,571,587,593,599,601,607,641,643,647,653,659

%N Ramanujan primes R_n: a(n) is the smallest number such that if x >= a(n), then pi(x) - pi(x/2) >= n, where pi(x) is the number of primes <= x.

%C Referring to his proof of Bertrand's postulate, Ramanujan states a generalization: "From this we easily deduce that pi(x) - pi(x/2) >= 1, 2, 3, 4, 5, ..., if x >= 2, 11, 17, 29, 41, ..., respectively." Since the a(n) are prime (by their minimality), I call them "Ramanujan primes."

%C 2n log 2n < a(n) < 4n log 4n for n >= 1, and Prime(2n) < a(n) < Prime(4n) if n > 1. Also, a(n) ~ Prime(2n) as n -> infinity.

%C Shanta Laishram has proved that a(n) < Prime(3n) for all n >= 1.

%C a(n) - 3n log 3n is sometimes positive, but negative with increasing frequency as n grows since a(n) ~ 2n log 2n. There should be a constant m s.t. for n >= m we have a(n) < 3n log 3n.

%C A good approximation to a(n) = R_n for n in [1..1000] is A162996(n) = Round(kn * (ln(kn)+1)), with k = 2.216 determined empirically from the first 1000 Ramanujan primes, which approximates the {kn}_th prime number which in turn approximates the n-th Ramanujan prime and where Abs(A162996(n) - R_n) < 2 * Sqrt(A162996(n)) for n in [1..1000]. Since R_n ~ Prime(2n) ~ 2n * (ln(2n)+1) ~ 2n * ln(2n), while A162996(n) ~ Prime(kn) ~ kn * (ln(kn)+1) ~ kn * ln(kn), A162996(n) / R_n ~ k/2 = 2.216/2 = 1.108 which implies an asymptotic overestimate of about 10% (a better approximation would need k to depend on n and be asymptotic to 2.) [_Daniel Forgues_, Jul 29 2009]

%C Let p_n be the n-th prime. If p_n>=3 is in the sequence, then all integers (p_n+1)/2, (p_n+3)/2, ..., (p_(n+1)-1)/2 are composite numbers. [_Vladimir Shevelev_, Aug 12 2009]

%C Denote by q(n) the prime which is the nearest from the right to a(n)/2. Then there exists a prime between a(n) and 2q(n). Converse, generally speaking, is not true, i.e. there exist primes outside the sequence, but possess such property (e.g., 109) [_Vladimir Shevelev_, Aug 14 2009]

%C The Mathematica program FasterRamanujanPrimeList uses Laishram's result that a(n) < Prime(3n).

%C See sequence A164952 for a generalization we call a Ramanujan k-prime. [_Vladimir Shevelev_, Sep 01 2009]

%C Contribution from _Jonathan Sondow_, May 22 2010: (Start)

%C About 46% of primes < 19000 are Ramanujan primes. About 78% of the lesser of twin primes < 19000 are Ramanujan primes.

%C About 15% of primes < 19000 are the lesser of twin primes. About 26% of Ramanujan primes < 19000 are the lesser of twin primes.

%C A reason for the jumps is in Section 7 of "Ramanujan primes and Bertrand's postulate" and in Section 4 of "Ramanujan Primes: Bounds, Runs, Twins, and Gaps".

%C See Shapiro 2008 for an exposition of Ramanujan's proof of his generalization of Bertrand's postulate. (End)

%C The 10^n_th R prime: 2, 97, 1439, 19403, 242057, 2916539, 34072993, 389433437,.... - _Robert G. Wilson v_, May 07 2011 & updated Aug 02 2012

%C The number of R_primes < 10^n: 1, 10, 72, 559, 4459, 36960, 316066, 2760321,.... - _Robert G. Wilson v_, Aug 02 2012

%C a(n) = R_n = R_{0.5,n} in "Generalized Ramanujan Primes."

%C All Ramanujan primes are in A164368. - _Vladimir Shevelev_, Aug 30 2011

%C If n tends to infinity, then limsup(a(n)-A080359(n-1)) = infty; conjecture: also limsup(a(n)-A080359(n)) = infty (cf. A182366). - _Vladimir Shevelev_, Apr 27 2012

%C Or the largest prime x such that the number of primes in (x/2,x] equals n. This equivalent definition underlines an important analogy between Ramanujan and Labos primes (cf. A080359). - _Vladimir Shevelev_, Apr 29 2012

%D Shanta Laishram, On a conjecture on Ramanujan primes, Int. J. Number Theory 6 (2010), 1869-1873.

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

%D S. Ramanujan, Collected Papers of Srinivasa Ramanujan (Ed. G. H. Hardy, S. Aiyar, P. Venkatesvara and B. M. Wilson), Amer. Math. Soc., Providence, 2000, pp. 208-209.

%D H. N. Shapiro, Ramanujan's idea, Section 9.3B in Introduction to the Theory of Numbers, Dover, 2008.

%D J. Sondow, Ramanujan primes and Bertrand's postulate, Amer. Math. Monthly, 116 (2009) 630-635.

%H T. D. Noe, <a href="/A104272/b104272.txt">Table of n, a(n) for n = 1..10000</a>

%H N. Amersi, O. Beckwith, S. J. Miller, R. Ronan, J. Sondow, <a href="http://arxiv.org/abs/1108.0475">Generalized Ramanujan Primes</a>, arXiv 2011.

%H Peter Hegarty, <a href="http://arxiv.org/abs/1201.3847">Why should one expect to find long runs of (non)-Ramanujan primes?</a>, arXiv 2012.

%H Shanta Laishram, <a href="http://www.isid.ac.in/~shanta/PAPERS/RamanujanPrimesIJNT.pdf">On a conjecture on Ramanujan primes</a>, 2010.

%H M. B. Paksoy, <a href="http://arxiv.org/abs/1210.6991">Derived Ramanujan primes: R'_n</a>, arXiv 2012.

%H PlanetMath, <a href="http://planetmath.org/ramanujanprime">Ramanujan prime</a>

%H S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper24/page1.htm">A Proof Of Bertrand's Postulate</a>, 1919.

%H V. Shevelev, <a href="http://arXiv.org/abs/0908.2319">On critical small intervals containing primes</a>

%H V. Shevelev, <a href="http://arxiv.org/abs/0909.0715">Ramanujan and Labos primes, their generalizations and classifications of primes</a>, arXiv 2009.

%H V. Shevelev, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL15/Shevelev/shevelev19.html">Ramanujan and Labos primes, their generalizations, and classifications of primes</a>, J. Integer Seq. 15 (2012) Article 12.5.4

%H V. Shevelev, C. R. Greathouse IV and P. J. C. Moses, <a href="http://arxiv.org/abs/1212.2785">On intervals (kn, (k+1)n) containing a prime for all n>1</a>, arXiv 2012.

%H J. Sondow, <a href="http://arxiv.org/abs/0907.5232">Ramanujan primes and Bertrand's postulate</a>, arXiv 2009.

%H J. Sondow, J. 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 J. Sondow, <a href="http://mathworld.wolfram.com/RamanujanPrime.html">Ramanujan Prime</a>, Eric Weisstein's MathWorld.

%H J. Sondow and E. Weisstein, <a href="http://mathworld.wolfram.com/BertrandsPostulate.html">Bertrand's Postulate</a>, MathWorld.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Ramanujan_prime">Ramanujan prime</a>

%F a(n) = 1 + max{k: pi(k) - pi(k/2) = n - 1}.

%F a(n) = A080360(n-1) + 1 for n > 1.

%F a(n) >= A080359(n). [_Vladimir Shevelev_, Aug 20 2009]

%F A193761(n) <= a(n) <= A193880(n).

%F a(n) = 2*A084140(n) - 1, for n > 1. - _Jonathan Sondow_, Dec 21 2012

%e a(1) = 2 is Bertrand's postulate: pi(x) - pi(x/2) >= 1 for all x >= 2.

%e a(2) = 11 because a(2) < 8 log 8 < 17 and pi(n) - pi(n/2) > 1 for n = 16, 15, ..., 11 but pi(10) - pi(5) = 1.

%e Consider a(9)=71. Then the nearest prime > 71/2 is 37, and between a(9) and 2*37, that is, between 71 and 74, there exists a prime (73). [Vladimir Shevelev, Aug 14 2009] - corrected by _Jonathan Sondow_, Jun 17 2013

%t (RamanujanPrimeList[n_] := With[{T=Table[{k,PrimePi[k]-PrimePi[k/2]}, {k,Ceiling[N[4*n*Log[4*n]]]}]}, Table[1+First[Last[Select[T,Last[ # ]==i-1&]]],{i,1,n}]]; RamanujanPrimeList[54]) (* _Jonathan Sondow_, Aug 15 2009 *)

%t (FasterRamanujanPrimeList[n_] := With[{T=Table[{k,PrimePi[k]-PrimePi[k/2]}, {k,Prime[3*n]}]}, Table[1+First[Last[Select[T,Last[ # ]==i-1&]]],{i,1,n}]]; FasterRamanujanPrimeList[54])

%t nn=1000; 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]}]; R=R+1 (* _T. D. Noe_, Nov 15 2010 *)

%Y Cf. A006992 Bertrand primes, A056171 pi(n) - pi(n/2).

%Y Cf. A000720, A014085, A060715, A084139, A084140, A143223, A143224, A143225, A143226, A143227, A080360, A080359, A164368, A164288, A164554, A164333, A164294, A164371.

%Y Cf. A162996 Round(kn * (ln(kn)+1)), with k = 2.216 as an approximation of R_n = n-th Ramanujan Prime.

%Y Cf. A163160 Round(kn * (ln(kn)+1)) - R_n, where k = 2.216 and R_n = n-th Ramanujan prime.

%Y Cf. A178127 Lesser of twin Ramanujan primes, A178128 Lesser of twin primes if it is a Ramanujan prime.

%Y Cf. A181671 (number of Ramanujan primes less than 10^n).

%Y Cf. A174635 (non-Ramanujan primes), A174602, A174641 (runs of Ramanujan and non-Ramanujan primes).

%Y Cf. A189993, A189994 (lengths of longest runs of Ramanujan and non-Ramanujan primes < 10^n).

%Y Cf. A190124 (constant of summation: 1/a(n)^2).

%Y Cf. A192820 (2- or derived Ramanujan primes R'_n), A192821, A192822, A192823, A192824, A225907.

%Y Cf. A193761 (0.25-Ramanujan primes), A193880 (0.75-Ramanujan primes).

%Y Cf. A212493, A212541.

%K nonn,nice,changed

%O 1,1

%A _Jonathan Sondow_, Feb 27 2005

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