A164554 -id:A164554 - cp's OEIS frontend

A182451 Numbers of A166252 which are not in A164554.

109, 151, 191, 229, 233, 283, 311, 571, 643, 683, 727, 941, 991, 1033, 1051, 1373, 1493, 1667, 1697, 1741, 1747, 1783, 1787, 1801, 1931

Offset: 1

Vladimir Shevelev, Apr 29 2012

nonn

All Ramanujan primes (A104272) are in A164368 and all Labos primes (A080359) are in A194598. Peculiar primes (see comment in A164554)are simultaneously Ramanujan and Labos primes, while central primes (A166252) are in the intersection of A164368 and A194598 for n>=2. Hence, for n>=2, all peculiar primes are central primes, but conversely is not true. The sequence lists non-peculiar central numbers.

Cf. A164554, A166252, A104272, A080359, A164368, A194598.

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.

Original entry on oeis.org

2, 11, 17, 29, 41, 47, 59, 67, 71, 97, 101, 107, 127, 149, 151, 167, 179, 181, 227, 229, 233, 239, 241, 263, 269, 281, 307, 311, 347, 349, 367, 373, 401, 409, 419, 431, 433, 439, 461, 487, 491, 503, 569, 571, 587, 593, 599, 601, 607, 641, 643, 647, 653, 659

Offset: 1

Jonathan Sondow, Feb 27 2005

Referring to his proof of Bertrand's postulate, Ramanujan (1919) 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 the arXiv link for a corrected version of Table 1.)
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)) = infinity; conjecture: also limsup(a(n) - A080359(n)) = infinity (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 proved for n > 10^300 by Shichun Yang and Alain Togbé in 2015. - Jonathan Sondow, Dec 01 2015
Berliner, Dean, Hook, Marr, Mbirika, and McBee (2016) prove in Theorem 18 that the graph K_{m,n} is prime for n >= R_{m-1}-m; see A291465. - Jonathan Sondow, May 21 2017
Okhotin (2012) uses Ramanujan primes to prove Lemma 8 in "Unambiguous finite automata over a unary alphabet." - Jonathan Sondow, May 30 2017
Sepulcre and Vidal (2016) apply Ramanujan primes in Remark 9 of "On the non-isolation of the real projections of the zeros of exponential polynomials." - Jonathan Sondow, May 30 2017
Axler and Leßmann (2017) compute the first k-Ramanujan prime for k >= 1 + epsilon; see A277718, A277719, A290394. - Jonathan Sondow, Jul 30 2017

			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]

Srinivasa 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.
Harold 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
N. Amersi, O. Beckwith, S. J. Miller, R. Ronan, and Jonathan Sondow, Generalized Ramanujan primes, arXiv:1108.0475 [math.NT], 2011.
N. Amersi, O. Beckwith, S. J. Miller, R. Ronan, and Jonathan 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. 39 (1) (2016) 1-30.
Christian Axler and Thomas Leßmann, An explicit upper bound for the first k-Ramanujan prime, arXiv:1504.05485 [math.NT], 2015.
Christian Axler and Thomas Leßmann, On the first k-Ramanujan prime, Amer. Math. Monthly, 124 (2017), 642-646.
Adam H. Berliner, N. Dean, J. Hook, A. Marr, A. Mbirika, and C. McBee, Coprime and prime labelings of graphs, arXiv preprint arXiv:1604.07698 [math.CO], 2016; Journal of Integer Sequences, Vol. 19 (2016), #16.5.8.
Paul 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.
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
Shanta Laishram, On a conjecture on Ramanujan primes, Int. J. Number Theory, 6 (2010), 1869-1873.
Catherine Lee, Minimum coprime graph labelings, arXiv:1907.12670 [math.CO], 2019.
Jaban Meher and M. Ram Murty, Ramanujan's proof of Bertrand's postulate, Amer. Math. Monthly, Vol. 120, No. 7 (2013), pp. 650-653.
Alexander Okhotin, Unambiguous finite automata over a unary alphabet, Inf. Comput., 212 (2012), 15-36.
Murat Baris Paksoy, Derived Ramanujan primes: R'_n, arXiv:1210.6991 [math.NT], 2012.
PlanetMath, Ramanujan prime
Srinivasa Ramanujan, A proof of Bertrand's postulate, J. Indian Math. Soc., 11 (1919), 181-182.
Juan Matias Sepulcre and Tomás Vidal, On the non-isolation of the real projections of the zeros of exponential polynomials, J. Math. Anal. Appl., 437 (2016) No. 1, 513-525.
Vladimir Shevelev, On critical small intervals containing primes, arXiv:0908.2319 [math.NT], 2009.
Vladimir Shevelev, Ramanujan and Labos primes, their generalizations and classifications of primes, arXiv:0909.0715 [math.NT], 2009-2011.
Vladimir 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, and 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
Jonathan Sondow, Ramanujan primes and Bertrand's postulate, arXiv:0907.5232 [math.NT], 2009-2010.
Jonathan Sondow, Ramanujan primes and Bertrand's postulate, Amer. Math. Monthly, 116 (2009), 630-635. Zentralblatt review.
Jonathan Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, arXiv:1105.2249 [math.NT] 2011; J. Integer Seq. 14 (2011) Article 11.6.2.
Jonathan Sondow, Ramanujan Prime, Eric Weisstein's MathWorld.
Jonathan Sondow and Eric 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.

Cf. A006992 (Bertrand primes), A056171 (pi(n) - pi(n/2)).
Cf. A000720, A007053, A014085, A060715, A084139, A084140, A143223, A143224, A143225, A143226, A143227, A080360, A080359, A164368, A164288, A164554, A164333, A164294, A164371, A190303.
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).
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, A277718, A277719, A164952, A290394, A291465.
Cf. A185004 - A185007 ("modular" Ramanujan primes).
Not to be confused with the Ramanujan numbers or Ramanujan tau function, A000594.

A104272 := proc(n::integer)
    local R;
    if n = 1 then
        return 2;
    end if;
    R := ithprime(3*n-1) ; # upper limit Laishram's thrm Thrm 3 arXiv:1105.2249
    while true do
        if A056171(R) = n then # Defn. 1. of Shevelev JIS 14 (2012) 12.1.1
            return R ;
        end if;
        R := prevprime(R) ;
    end do:
end proc:
seq(A104272(n),n=1..200) ; # slow downstream search <= p(3n-1) R. J. Mathar, Sep 21 2017

(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[sT. D. Noe, Nov 15 2010 *)

ramanujan_prime_list(n) = {my(L=vector(n), s=0, k=1); for(k=1, prime(3*n)-1, if(isprime(k), s++); if(k%2==0 && isprime(k/2), s--); if(sSatish Bysany, Mar 02 2017

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

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(n) = A000040(A179196(n)). - R. J. Mathar, Sep 21 2017
Sum_{n>=1} (-1)^(n+1)/a(n) = A190303. - Amiram Eldar, Nov 20 2020

A080359 The smallest integer x > 0 such that the number of primes in (x/2, x] equals n.

Original entry on oeis.org

2, 3, 13, 19, 31, 43, 53, 61, 71, 73, 101, 103, 109, 113, 139, 157, 173, 181, 191, 193, 199, 239, 241, 251, 269, 271, 283, 293, 313, 349, 353, 373, 379, 409, 419, 421, 433, 439, 443, 463, 491, 499, 509, 523, 577, 593, 599, 601, 607, 613, 619, 647, 653, 659

Offset: 1

Labos Elemer, Feb 21 2003

nonn

a(n) is the same as: Smallest integer x > 0 such that the number of unitary-prime-divisors of x! equals n.
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
For n >= 3, denote by q(n) the prime which is the nearest from the left to a(n)/2. Then there exists a prime between 2q(n) and a(n). The converse, generally speaking, is not true; i.e., there exist primes that are outside the sequence, but possess such property (e.g., 131). - Vladimir Shevelev, Aug 14 2009
See sequence A164958 for a generalization. - Vladimir Shevelev, Sep 02 2009
a(n) is the n-th Labos prime.

			n=5: in 31! five unitary-prime-divisors appear (firstly): {17,19,23,29,31}, while other primes {2,3,5,7,11,13} are at least squared. Thus a(5)=31.
Consider a(9)=71. Then the nearest prime < 71/2 is q(9)=31, and between 2q(9) and a(9), i.e., between 62 and 71 there exists a prime (67). - _Vladimir Shevelev_, Aug 14 2009

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..4460 from Daniel Forgues)
N. Amersi, O. Beckwith, S. J. Miller, R. Ronan, and J. Sondow, Generalized Ramanujan primes, arXiv:1108.0475 [math.NT], 2011.
N. Amersi, O. Beckwith, S. J. Miller, R. Ronan, and J. Sondow, Generalized Ramanujan primes, Combinatorial and Additive Number Theory, Springer Proc. in Math. & Stat., CANT 2011 and 2012, Vol. 101 (2014), 1-13.
Ethan Berkove and Michael Brilleslyper, Subgraphs of Coprime Graphs on Sets of Consecutive Integers, Integers (2022) Vol. 22, #A47, see p. 8.
Vladimir Shevelev, On critical small intervals containing primes, arXiv:0908.2319 [math.NT], 2009.
Vladimir Shevelev, Ramanujan and Labos primes, their generalizations and classifications of primes, arXiv:0909.0715 [math.NT], 2009-2011.
Vladimir Shevelev, Ramanujan and Labos primes, their generalizations, and classifications of primes, J. Integer Seq. 15 (2012) Article 12.5.4.
Jonathan Sondow, MathWorld: Ramanujan Prime
Jonathan Sondow, Ramanujan primes and Bertrand's postulate, arXiv:0907.5232 [math.NT], 2009-2010.
Jonathan Sondow, Ramanujan primes and Bertrand's postulate, Amer. Math. Monthly, 116 (2009), 630-635.

Cf. A056171, A056169, A000720, A000142.
Cf. A104272 (Ramanujan primes).
Cf. A060756, A080360 (largest integer x with n primes in (x/2,x]).
Cf. A164554, A164288, A164333, A164294, A164372, A164371, A212493, A212541.

nn=1000; t=Table[0, {nn+1}]; s=0; Do[If[PrimeQ[k], s++]; If[PrimeQ[k/2], s--]; If[s<=nn && t[[s+1]]==0, t[[s+1]]=k], {k, Prime[3*nn]}]; Rest[t]
(* Second program: *)
a[1] = 2; a[n_] := a[n] = Module[{x = a[n-1]}, While[(PrimePi[x]-PrimePi[Quotient[x, 2]]) != n, x++ ]; x]; Array[a, 54] (* Jean-François Alcover, Sep 14 2018 *)

a(n) = {my(x = 1); while ((primepi(x) - primepi(x\2)) != n, x++;); x;} \\ Michel Marcus, Jan 15 2014

def A():
    i = 0; n = 1
    while True:
        p = prime_pi(i) - prime_pi(i//2)
        if p == n:
            yield i
            n += 1
        i += 1
A080359 = A()
[next(A080359) for n in range(54)] # Peter Luschny, Sep 03 2014

a(n) = Min{x; Pi[x]-Pi[x/2]=n} = Min{x; A056171(x)=n}=Min{x; A056169(n!)=n}; where Pi()=A000720().

a(n) <= A193507(n) (cf. A194186). - Vladimir Shevelev, Aug 18 2011

Definition corrected by Jonathan Sondow, Aug 10 2008

Shrunk title and moved part of title to comments by John W. Nicholson, Sep 18 2011

A164333 Primes prime(k) such that all integers in the interval [(prime(k-1)+1)/2, (prime(k)-1)/2] are composite numbers.

Original entry on oeis.org

13, 19, 31, 43, 53, 61, 71, 73, 101, 103, 109, 113, 131, 139, 151, 157, 173, 181, 191, 193, 199, 229, 233, 239, 241, 251, 269, 271, 283, 293, 311, 313, 349, 353, 373, 379, 409, 419, 421, 433, 439, 443, 463, 491, 499, 509, 523, 571, 577, 593, 599, 601, 607, 613, 619, 643

Offset: 1

Vladimir Shevelev, Aug 13 2009

nonn

Let p_k be the k-th prime. A prime p is in the sequence iff the interval of the form (2p_k, 2p_(k+1)), containing p, also contains a prime less than p. The sequence is connected with the following classification of primes: the first two primes 2,3 form a separate set of primes; let p >= 5 be in the interval (2p_k, 2p_(k+1)), then 1) if in this interval there are only primes greater than p, then p is called a right prime; 2) if in this interval there are only primes less than p, then p is called a left prime; 3) if in this interval there are primes both greater and less than p, then p is called a central prime; 4) if this interval does not contain other primes, then p is called an isolated prime. In particular, the right primes form sequence A166307, and all Ramanujan primes (A104272) greater than 2 are either right or central primes; the left primes form sequence A182365, and all Labos primes (A080359) greater than 3 are either left or central primes; the central primes form A166252 and the isolated primes form A166251. [Vladimir Shevelev, Oct 10 2009] [Sequence reference updated by Peter Munn, Jun 01 2023]

Disjoint union of A166252 and A182365. - Peter Munn, Jun 01 2023 [an edited version of a contribution by Vladimir Shevelev in 2009]

			Let p=53. We see that 2*23<53<2*29. Since the interval (46, 58) contains prime 47<53 and does not contain any prime more than 53, then, by the considered classification 53 is left prime and it is in the sequence. [_Vladimir Shevelev_, Oct 10 2009]

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, J. Int. Seq. 15 (2012) # 12.5.4
J. Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, arXiv:1105.2249 [math.NT], 2011; J. Integer Seq. 14 (2011) Article 11.6.2.

Cf. A080359, A104272, A164288, A164294, A164332, A001262, A001567, A062568, A141232.
Cf. A164368, A164554, A166251, A166252. [Vladimir Shevelev, Oct 10 2009]
Cf. A166307, A182365.

isA164333 := proc(n)
        local i ;
        if isprime(n) and n > 3 then
                for i from (prevprime(n)+1)/2 to (n-1)/2 do
                        if isprime(i) then
                                return false;
                        end if;
                end do;
                return true;
        else
                false;
        end if;
end proc:
for i from 2 to 700 do
        if isA164333(i) then
                printf("%d,",i);
        end if;
end do: # R. J. Mathar, Oct 29 2011

kmax = 200; Select[Table[{(Prime[k - 1] + 1)/2, (Prime[k] - 1)/2}, {k, 3, kmax}], AllTrue[Range[#[[1]], #[[2]]], CompositeQ]&][[All, 2]]*2 + 1 (* Jean-François Alcover, Nov 14 2017 *)

{A080359} union {A164294} = {this sequence} union {2,3}. - Vladimir Shevelev, Oct 29 2011

A164368(2)A164368(3)A164368(4)Vladimir Shevelev, Oct 10 2009]

Definition rephrased by R. J. Mathar, Oct 02 2009

A166252 Primes which are not the smallest or largest prime in an interval of the form (2prime(k),2prime(k+1)).

Original entry on oeis.org

71, 101, 109, 151, 181, 191, 229, 233, 239, 241, 269, 283, 311, 349, 373, 409, 419, 433, 439, 491, 571, 593, 599, 601, 607, 643, 647, 653, 659, 683, 727, 823, 827, 857, 941, 947, 991, 1021, 1031, 1033, 1051, 1061, 1063, 1091, 1103, 1301, 1373, 1427, 1429

Offset: 1

Vladimir Shevelev, Oct 10 2009, Oct 14 2009

nonn

Called "central primes" in A166251, not to be confused with the central polygonal primes A055469.
The primes tabulated in intervals (2*prime(k),2*prime(k+1)) are
5, k=1
7, k=2
11,13, k=3
17,19, k=4
23,    k=5
29,31, k=6
37,    k=7
41,43, k=8
47,53, k=9
59,61, k=10
67,71,73,  k=11
79,        k=12
83,     k=13
89,     k=14
97,101,103, k=15
and only rows with at least 3 primes contribute primes to the current sequence.
For n >= 2, these are numbers of A164368 which are in A194598. - Vladimir Shevelev, Apr 27 2012

			Since 2*31 < 71 < 2*37 and the interval (62, 74) contains prime 67 < 71 and prime 73 > 71, then 71 is in the sequence.

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

Cf. A166307, A182365, A166251, A164368, A104272, A080359, A164333, A164288, A164294, A164554, A194598.

n = 0; t = {}; While[Length[t] < 100, n++; ps = Select[Range[2*Prime[n], 2*Prime[n+1]], PrimeQ]; If[Length[ps] > 2, t = Join[t, Rest[Most[ps]]]]]; t (* T. D. Noe, Apr 30 2012 *)

A166307 The smallest prime in some interval of the form (2prime(k),2prime(k+1)) if this interval contains at least 2 primes.

Original entry on oeis.org

11, 17, 29, 41, 47, 59, 67, 97, 107, 127, 137, 149, 167, 179, 197, 227, 263, 281, 307, 347, 367, 401, 431, 461, 487, 503, 521, 569, 587, 617, 641, 677, 719, 739, 751, 769, 809, 821, 853, 881, 907, 937, 967, 983, 1009, 1019, 1049, 1087, 1097, 1117, 1151, 1163, 1187, 1217, 1229, 1249, 1277

Offset: 1

Vladimir Shevelev, Oct 11 2009, Oct 17 2009

nonn

These are called "right primes" in A166251.

			For p=29 we have: 2*13 < 29 < 2*17 and interval (26, 29) is free from primes while interval (29, 34) contains a prime. Therefore 29 is in the sequence for k=6.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A166252, A166251, A164368, A104272, A080359, A164333, A164288, A164294, A164554, A182365.

f[n_] := Block[{t = Select[ Table[i, {i, 2 Prime[n], 2 Prime[n + 1]}], PrimeQ]}, If[ Length@ t > 1, t[[1]], 0]]; Rest@ Union@ Array[f, 115] (* Robert G. Wilson v, May 08 2011 *)

A164371 The smaller members of twin prime pairs which are non-Ramanujan primes.

Original entry on oeis.org

3, 5, 137, 191, 197, 521, 617, 1319, 1619, 1697, 1931, 1949, 2111, 2549, 2591, 2969, 3119, 3329, 3389, 3557, 3929, 4157, 4217, 4271, 4517, 5279, 5417, 7331, 7349, 7757, 8009, 8429, 8837, 9011, 9281, 9437, 9857, 9929, 10007, 10037, 10457, 10529, 10859

Offset: 1

Vladimir Shevelev, Aug 14 2009, Aug 17 2009

nonn

The terms greater than 5 of the sequence are in A164288.

Dana Jacobsen, Table of n, a(n) for n = 1..10000
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

Cf. A104272, A001359, A164288, A164554

nn = 1000;
A001359 = Select[Prime[Range[2 nn]], PrimeQ[# + 2] &];
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]}];
A104272 = R + 1;
Complement[A001359, A104272] (* Jean-François Alcover, Oct 28 2018, after T. D. Noe in A104272 *)

use ntheory ":all"; for (@{twin_primes(1e4)}) { say if !is_ramanujan_prime($) } # _Dana Jacobsen, Dec 30 2015

A001359 \ A104272. - R. J. Mathar, Sep 17 2009

Extended by R. J. Mathar, Sep 17 2009

A182365 The largest prime in some interval of the form (2prime(k),2prime(k+1)) if this interval contains at least 2 primes.

Original entry on oeis.org

13, 19, 31, 43, 53, 61, 73, 103, 113, 131, 139, 157, 173, 193, 199, 251, 271, 293, 313, 353, 379, 421, 443, 463, 499, 509, 523, 577, 613, 619, 661, 691, 733, 743, 757, 773, 811, 829, 859, 883, 911, 953, 971, 997, 1013, 1039, 1069, 1093, 1109, 1123, 1153

Offset: 1

Vladimir Shevelev, Apr 26 2012

nonn

These are called "left primes" in A166251.

			For k=6 we have 2*13 < 29 < 31 < 2*17, and the interval contains two primes. Therefore 31 is in the sequence.

Cf. A166252, A166251, A164368, A104272, A080359, A164333, A164288, A164294, A164554, A166307.

n = 0; t = {}; While[Length[t] < 100, n++; ps = Select[Range[2*Prime[n], 2*Prime[n + 1]], PrimeQ]; If[Length[ps] >= 2, AppendTo[t, ps[[-1]]]]]; t (* T. D. Noe, Apr 30 2012 *)

A212493 Let p_n=prime(n), n>=1. Then a(n) is the least prime p which differs from p_n, for which the intervals (p/2,p_n/2), (p,p_n], if pp_n, contain the same number of primes, and a(n)=0, if no such prime p exists.

Original entry on oeis.org

0, 5, 3, 3, 3, 17, 13, 23, 19, 19, 37, 31, 31, 47, 43, 59, 53, 67, 61, 0, 79, 73, 73, 73, 73, 0, 107, 103, 127, 131, 109, 113, 113, 151, 113, 139, 163, 157, 157, 179, 173, 0, 223, 197, 193, 233, 193, 191, 191, 193, 199, 0, 0, 257, 251, 251, 0, 277, 271, 271

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, May 18 2012

nonn

a(n)=0 if and only if p_n is a peculiar prime, i.e., simultaneously Ramanujan (A104272) and Labos (A080359) prime (see sequence A164554).

a(n)>p_n if and only if p_n is Labos prime but not Ramanujan prime.

			Let n=5, p_5=11; p=2 is not suitable, since in (1,5.5) we have 3 primes, while in (2,11] there are 4 primes. Consider p=3. Now in intervals (1.5,5.5) and (3,11] we have the same number (3) of primes. Therefore, a(5)=3. The same value we can obtain by the formula. Since 11 is not a Labos prime, then a(5)=A080359(5-pi(5.5))=A080359(2)=3.

Vladimir Shevelev, Ramanujan and Labos primes, their generalizations, and classifications of primes, J. Integer Seq. 15 (2012) Article 12.5.4.
Jonathan Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, arXiv:1105.2249 [math.NT], 2011; J. Integer Seq. 14 (2011) Article 11.6.2.

Cf. A104272, A080359, A164554.

terms = 60; nn = Prime[terms];
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]}];
A104272 = R + 1;
t = Table[0, {nn + 1}]; s = 0; Do[If[PrimeQ[k], s++]; If[PrimeQ[k/2], s--]; If[s <= nn && t[[s + 1]] == 0, t[[s + 1]] = k], {k, Prime[3 nn]}];
A080359 = Rest[t];
a[n_] := Module[{}, pn = Prime[n]; If[MemberQ[A104272, pn] && MemberQ[ A080359, pn], Return[0]]; For[p = 2, True, p = NextPrime[p], Which[ppn, If[PrimePi[p/2] - PrimePi[pn/2] == PrimePi[p] - PrimePi[pn], Return[p]]]]];
Array[a, terms] (* Jean-François Alcover, Dec 04 2018, after T. D. Noe in A104272 *)

If p_n is not a Labos prime, then a(n) = A080359(n-pi(p_n/2)).

A212541 Let p_n=prime(n), n>=1. Then a(n) is the maximal prime p which differs from p_n, for which the intervals (p/2,p_n/2), (p,p_n], if pp_n, contain the same number of primes, and a(n)=0, if no such prime p exists.

Original entry on oeis.org

0, 11, 11, 11, 7, 17, 13, 29, 29, 23, 41, 41, 37, 47, 43, 59, 53, 67, 61, 0, 97, 97, 97, 97, 89, 0, 107, 103, 127, 149, 109, 149, 149, 151, 137, 139, 167, 167, 163, 179, 173, 0, 227, 229, 229, 233, 229, 227, 223, 211, 199, 0, 0, 263, 263, 257, 0, 281, 281

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, May 20 2012

nonn

a(n)A104272).

a(n)=0 if and only if p_n is a peculiar prime, i.e., simultaneously Ramanujan and Labos (A080359) prime (see sequence A164554).

			Let n=4, p_n=7. Since 7 is not Ramanujan prime, then a(4) = A104272(4-pi(3.5)) = A104272(2) = 11.

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

Cf. A212493, A104272, A080359, A164554.

If p_n is not a Ramanujan prime, then a(n) = A104272(n-pi(p_n/2)).

cp's OEIS Frontend

A182451 Numbers of A166252 which are not in A164554.

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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.

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A080359 The smallest integer x > 0 such that the number of primes in (x/2, x] equals n.

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A164333 Primes prime(k) such that all integers in the interval [(prime(k-1)+1)/2, (prime(k)-1)/2] are composite numbers.

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A166252 Primes which are not the smallest or largest prime in an interval of the form (2*prime(k),2*prime(k+1)).

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A166307 The smallest prime in some interval of the form (2*prime(k),2*prime(k+1)) if this interval contains at least 2 primes.

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A164371 The smaller members of twin prime pairs which are non-Ramanujan primes.

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A166252 Primes which are not the smallest or largest prime in an interval of the form (2prime(k),2prime(k+1)).

A166307 The smallest prime in some interval of the form (2prime(k),2prime(k+1)) if this interval contains at least 2 primes.

A182365 The largest prime in some interval of the form (2prime(k),2prime(k+1)) if this interval contains at least 2 primes.