1,1

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."

See the additional references and links mentioned in A143227.

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.

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

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 such that for n >= m we have a(n) < 3n log 3n.

A good approximation to a(n) = R_n for n in [1..1000] is A162996(n) = Round(k*n * (log(k*n)+1)), with k = 2.216 determined empirically from the first 1000 Ramanujan primes, which approximates the {k*n}-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 * (log(2n)+1) ~ 2n * log(2n), while A162996(n) ~ prime(k*n) ~ k*n * (log(k*n)+1) ~ k*n * log(k*n), 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

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

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

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

See sequence A164952 for a generalization we call a Ramanujan k-prime. - Vladimir Shevelev, Sep 01 2009

From Jonathan Sondow, May 22 2010: (Start)

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

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

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".

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

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

The number of R primes < 10^n: 1, 10, 72, 559, 4459, 36960, 316066, 2760321, .... - Robert G. Wilson v, Aug 02 2012

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

All Ramanujan primes are in A164368. - Vladimir Shevelev, Aug 30 2011

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

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

Research questions on R_n - prime(2n) are at A233739, and on n-Ramanujan primes at A225907. - Jonathan Sondow, Dec 16 2013

The questions on R_n - prime(2n) in A233739 have been answered by Christian Axler in "On generalized Ramanujan primes". - Jonathan Sondow, Feb 13 2014

Srinivasan's Lemma (2014): prime(k-n) < prime(k)/2 if R_n = prime(k) and n > 1. Proof: By the minimality of R_n, the interval (prime(k)/2,prime(k)] contains exactly n primes and so prime(k-n) < prime(k)/2. - Jonathan Sondow, May 10 2014

For some n and k, we see that A168421(k) = a(n) so as to form a chain of primes similar to a Cunningham chain. For example (and the first example), A168421(2) = 7, links a(2) = 11 = A168421(3), links a(3) = 17 = A168421(4), links a(4) = 29 = A168421(6), links a(6) = 47. Note that the links do not have to be of a form like q = 2*p+1 or q = 2*p-1. - John W. Nicholson, Feb 22 2015

Extending Sondow's 2010 comments: About 48% of primes < 10^9 are Ramanujan primes.  About 76% of the lesser of twin primes < 10^9 are Ramanujan primes. - Dana Jacobsen, Sep 06 2015

Sondow, Nicholson, and Noe's 2011 conjecture that pi(R_{m*n}) <= m*pi(R_n) for m >= 1 and n >= N_m (see A190413, A190414) was proven for n > 10^300 by Shichun Yang and Alain Togbé in 2015. - Jonathan Sondow, Dec 01 2015

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.

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

T. D. Noe, Table of n, a(n) for n = 1..10000

Sequences related to Ramanujan primes

N. Amersi, O. Beckwith, S. J. Miller, R. Ronan, J. Sondow, Generalized Ramanujan primes, arXiv:1108.0475 [math.NT], 2011.

N. Amersi, O. Beckwith, S. J. Miller, R. Ronan, J. Sondow, Generalized Ramanujan primes, Combinatorial and Additive Number Theory, Springer Proc. in Math. & Stat., CANT 2011 and 2012, Vol. 101 (2014), 1-13

Christian Axler, Über die Primzahl-Zählfunktion, die n-te Primzahl und verallgemeinerte Ramanujan-Primzahlen, Ph.D. thesis 2013, in German, English summary.

Christian Axler, On generalized Ramanujan primes, arXiv:1401.7179 [math.NT], 2014.

Christian Axler, On generalized Ramanujan primes, Ramanujan J., online 30 April 2015, 1-30.

Christian Axler and Thomas Leßmann, An explicit upper bound for the first k-Ramanujan prime, arXiv:1504.05485 [math.NT], 2015.

P. Erdős, A theorem of Sylvester and Schur, J. London Math. Soc., 9 (1934), 282-288.

Peter Hegarty, Why should one expect to find long runs of (non)-Ramanujan primes?, arXiv:1201.3847 [math.NT], 2012.

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

Jaban Meher and M. Ram Murty, Ramanujan’s proof of Bertrand’s postulate, Amer. Math. Monthly, 120 (2013), 650-653.

M. B. Paksoy, Derived Ramanujan primes: R'_n, arXiv:1210.6991 [math.NT], 2012.

PlanetMath, Ramanujan prime

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

V. Shevelev, On critical small intervals containing primes, arXiv:0908.2319 [math.NT], 2009.

V. Shevelev, Ramanujan and Labos primes, their generalizations and classifications of primes, arXiv:0909.0715 [math.NT], 2009, 2013.

V. Shevelev, Ramanujan and Labos primes, their generalizations, and classifications of primes, J. Integer Seq. 15 (2012) Article 12.5.4

Vladimir Shevelev, Charles R. Greathouse IV, Peter J. C. Moses, On intervals (kn, (k+1)n) containing a prime for all n>1, Journal of Integer Sequences, Vol. 16 (2013), Article 13.7.3. arXiv:1212.2785

J. Sondow, Ramanujan primes and Bertrand's postulate, arXiv:0907.5232 [math.NT], 2009-2010.

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

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, Eric Weisstein's MathWorld.

J. Sondow and E. Weisstein, Bertrand's Postulate, MathWorld.

Anitha Srinivasan, An upper bound for Ramanujan primes, Integers, 14 (2014), #A19.

Anitha Srinivasan and John W. Nicholson, An improved upper bound for Ramanujan primes, Integers, 15 (2015), #A52.

Wikipedia, Bertrand's postulate

Wikipedia, Ramanujan prime

Shichun Yang and Alain Togbé, On the estimates of the upper and lower bounds of Ramanujan primes, Ramanujan J., online 14 August 2015, 1-11.

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

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

a(n) >= A080359(n). - Vladimir Shevelev, Aug 20 2009

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

a(n) = 2*A084140(n) - 1, for n > 1. - Jonathan Sondow, Dec 21 2012

a(n) = prime(2n) + A233739(n) = (A233822(n) + a(n+1))/2. - Jonathan Sondow, Dec 16 2013

a(n) = max{prime p: pi(p) - pi(p/2) = n} (see Shevelev 2012). - Jonathan Sondow, Mar 23 2016

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

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.

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

(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 *)

(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])

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 *)

(Perl) use ntheory ":all"; my $r = ramanujan_primes(1000); say "[@$r]"; # Dana Jacobsen, Sep 06 2015

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

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

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

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

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

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

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

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

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

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

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

Cf. A190413, A190414, A212493, A212541, A233739, A233822.

Not to be confused with the Ramanujan numbers or Ramanujan tau function, A000594.

Sequence in context: A019364 A164368 A194658 * A214934 A233866 A117155

Adjacent sequences:  A104269 A104270 A104271 * A104273 A104274 A104275

nonn,nice

Jonathan Sondow, Feb 27 2005

approved