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. [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 03 2008]

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. [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Apr 26 2009]

Shanta Laishram has proved that a(n) < Prime(3n) for all n >= 1. [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Jun 12 2009]

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.

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.) [From Daniel Forgues (squid(AT)zensearch.com), 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. [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), 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) [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Aug 14 2009]

The Mathematica program FasterRamanujanPrimeList uses Laishram's result that a(n) < Prime(3n). [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 15 2009]

A generalization. For k>1 (not necessarily integer), we call a Ramanujan k-prime R_n^(k) the prime a_k(n) which is the smallest number such that if x >= a_k(n), then pi(x)- pi(x/k) >= n. Note that, the sequence of all primes corresponds to the case of "k=oo". These numbers possess the following properties: R_n^(k)~p_((k/(k-1))n) as n tends to the infinity; if A_k(x) is the counting function of the Ramanujan k-primes not exceeding x, then A_k(x)~(1-1/k)\pi(x); let p be a k-Ramanujan prime, such that p/k is in the interval (p_m, p_(m+1)), where p_m>=3 and p_n is the nth prime. Then the interval (p, k*p_(m+1)) contains a prime. Conjecture. For every k>=2 there exist non-k-Ramanujan primes, which possess the latter property. For example, for k=2, the smallest such prime is 109. Problem. For every k>2 to estimate the smallest non-k-Ramanujan prime,which possess the latter property. [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Sep 01 2009]

Shanta Laishram, On a conjecture on Ramanujan primes, preprint, 2009. [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Jun 12 2009]

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

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.

J. Sondow, Ramanujan primes and Bertrand's postulate, Amer. Math.���Monthly, 116 (2009) 630-635. [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Apr 26 2009]

T. D. Noe, Table of n, a(n) for n=1..1000

S. Ramanujan, A Proof Of Bertrand's Postulate

V. Shevelev, On critical small intervals containing primes [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Aug 20 2009]

J. Sondow, Ramanujan primes and Bertrand's postulate [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Apr 26 2009]

Eric Weisstein's World of Mathematics, Bertrand's Postulate

Eric Weisstein's World of Mathematics, Ramanujan Prime

Wikipedia, Ramanujan prime

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

a(n) = A080360(n-1) + 1 for n > 1 [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 11 2008]

a(n)>=A080359(n). [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Aug 20 2009]

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. [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Apr 26 2009]

Consider a(9)=71. Then the nearest prime>71/2 is q(9)=37, and between a(9) and 2q(9), i.e. between 71 and 74 there exists a prime (73). [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Aug 14 2009]

(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]) [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), 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]) [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 15 2009]

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

Cf. A000720, A014085, A060715, A143223, A143224, A143225, A143226, A143227. [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 03 2008]

Cf. A080360. [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 11 2008]

Contribution from Daniel Forgues (squid(AT)zensearch.com), Jul 21 2009: (Start)

Cf. A162996 Round(kn * (ln(kn)+1)), with k = 2.216 as an approximation of R_n = n_th Ramanujan Prime.

Cf. A163160 Round(kn * (ln(kn)+1)) - R_n, where k = 2.216 and R_n = n_th Ramanujan prime. (End)

A080359 A164368 A164288 A164554 A164333 A164294 A164371 [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Aug 20 2009]

Sequence in context: A087379 A019364 A164368 this_sequence A117155 A141176 A118839

Adjacent sequences: A104269 A104270 A104271 this_sequence A104273 A104274 A104275

nonn,nice

Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Feb 27 2005

Link corrected by Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Jul 31 2009