A104272 -id:A104272 - cp's OEIS frontend

A194658 a(n) is the maximal prime, such that for all primes x<=a(n) the number of primes in (x/2,x) is less than n.

2, 11, 17, 29, 41, 47, 59, 67, 71, 97, 101, 107, 109, 137, 151, 167, 179, 181, 191, 197, 233, 239, 241, 263, 269, 281, 283, 311, 347, 349, 367, 373, 401, 409, 419, 431, 433, 439, 461, 487, 491, 503, 521, 571, 587, 593, 599, 601, 607, 617, 643, 647, 653

Offset: 1

Vladimir Shevelev, Sep 01 2011

nonn

The next prime after a(n) is A080359(n+1).

Alois P. Heinz, Table of n, a(n) for n = 1..1000
V. Shevelev, Ramanujan and Labos primes, their generalizations and classifications of primes, arXiv:0909.0715 [math.NT], 2009-2011.
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.

Subsequence of A164368.

Cf. A080359, A193507, A104272.

b[1] = 2; b[n_] := b[n] = Module[{x = b[n-1]}, While[PrimePi[x] - PrimePi[ Quotient[x, 2]] != n, x++]; x];
a[n_] := NextPrime[b[n+1], -1];
Array[a, 100] (* Jean-François Alcover, Nov 11 2018 *)

A080359(n) <= a(n) <= A104272(n).

A143223 (Number of primes between n^2 and (n+1)^2) - (number of primes between n and 2n).

Original entry on oeis.org

0, 2, 1, 1, 1, 1, 2, 1, 2, 0, 1, 1, 1, 2, 1, 2, 2, 1, 2, 2, 3, 2, 1, 1, 3, 2, 1, 1, 2, 2, 1, 3, 2, 3, 1, 2, 0, 0, 3, 2, 2, 2, -1, 3, 2, 3, 0, 4, 6, 0, 1, 4, 4, 1, 1, -2, -1, 3, -1, 3, 3, 1, 5, 3, 1, 3, 1, 2, 4, -1, 6, 1, 1, 4, 4, 4, 7, -1, 3, 8, -2, 5, 3, 5, 1, 0, 5, 5, 1, 2, 3, 2, 1, 5, 3, 3, 2, 3, 4, 1, 2

Offset: 0

Jonathan Sondow, Jul 31 2008

sign

Legendre's conjecture (still open) says there is always a prime between n^2 and (n+1)^2. Bertrand's postulate (actually a theorem due to Chebyshev) says there is always a prime between n and 2n.
Hashimoto's plot of (1 - a(n)) shows that |a(n)| is small compared to n for n < 30000.
From Jonathan Sondow, Aug 07 2008: (Start)
It appears that there are only a finite number of negative terms (see A143226).
If the negative terms are bounded, then Legendre's conjecture is true, at least for all sufficiently large n. This follows from the strong form of Bertrand's postulate proved by Ramanujan (see A104272 Ramanujan primes). (End)

			There are 4 primes between 6^2 and 7^2 and 2 primes between 6 and 2*6, so a(6) = 4 - 2 = 2.
a(1) = 2 because there are two primes between 1^2 and 2^2 (namely, 2 and 3) and none between 1 and 2. [_Jonathan Sondow_, Aug 07 2008]

M. Aigner and C. M. Ziegler, Proofs from The Book, Chapter 2, Springer, NY, 2001.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 5th ed., Oxford Univ. Press, 1989, p. 19.
S. Ramanujan, Collected Papers of Srinivasa Ramanujan (G. H. Hardy, S. Aiyar, P. Venkatesvara and B. M. Wilson, eds.), Amer. Math. Soc., Providence, 2000, pp. 208-209.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
T. Hashimoto, On a certain relation between Legendre's conjecture and Bertrand's postulate, arXiv:0807.3690 [math.GM], 2008.
M. Hassani, Counting primes in the interval (n^2,(n+1)^2), arXiv:math/0607096 [math.NT], 2006.
T. D. Noe, Plot of A143223(n) for n to 10^6
J. Pintz, Landau's problems on primes
S. Ramanujan, A proof of Bertrand's postulate, J. Indian Math. Soc., 11 (1919), 181-182.
J. Sondow and E. W. Weisstein, Bertrand's Postulate in MathWorld
J. Sondow, Ramanujan Prime in MathWorld
E. W. Weisstein, Legendre's Conjecture in MathWorld

See A000720, A014085, A060715, A143224, A143225, A143226.

Negative terms are A143227. Cf. A104272 (Ramanujan primes).

L={0,2}; Do[L=Append[L,(PrimePi[(n+1)^2]-PrimePi[n^2]) - (PrimePi[2n]-PrimePi[n])], {n,2,100}]; L

a(n)=sum(k=n^2+1,n^2+2*n,isprime(k))-sum(k=n+1,2*n,isprime(k)) \\ Charles R Greathouse IV, May 30 2014

a(n) = A014085(n) - A060715(n) (for n > 0) = [pi((n+1)^2) - pi(n^2)] - [pi(2n) - pi(n)] (for n > 1).

Corrected by Jonathan Sondow, Aug 07 2008, Aug 09 2008

A143224 Numbers n such that (number of primes between n^2 and (n+1)^2) = (number of primes between n and 2n).

Original entry on oeis.org

0, 9, 36, 37, 46, 49, 85, 102, 107, 118, 122, 127, 129, 140, 157, 184, 194, 216, 228, 360, 365, 377, 378, 406, 416, 487, 511, 571, 609, 614, 672, 733, 767, 806, 813, 863, 869, 916, 923, 950, 978, 988, 1249, 1279, 1280, 1385, 1427, 1437, 1483, 1539, 1551, 1690

Offset: 1

Jonathan Sondow, Jul 31 2008

nonn

The sequence gives the positions of zeros in A143223. The number of primes in question is A143225(n).

Legendre's conjecture (still open) says there is always a prime between n^2 and (n+1)^2. Bertrand's postulate (actually a theorem due to Chebyshev) says there is always a prime between n and 2n.

			There is the same number of primes (namely 3) between 9^2 and 10^2 as between 9 and 2*9, so 9 is a term.

M. Aigner and C. M. Ziegler, Proofs from The Book, Chapter 2, Springer, NY, 2001.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 5th ed., Oxford Univ. Press, 1989, p. 19.
S. Ramanujan, Collected Papers of Srinivasa Ramanujan (G. H. Hardy, S. Aiyar, P. Venkatesvara and B. M. Wilson, eds.), Amer. Math. Soc., Providence, 2000, pp. 208-209. [Jonathan Sondow, Aug 03 2008]

T. D. Noe, Table of n, a(n) for n=1..97 (no other n < 10^6)
T. Hashimoto, On a certain relation between Legendre's conjecture and Bertrand's postulate, arXiv:0807.3690 [math.GM], 2008.
M. Hassani, Counting primes in the interval (n^2,(n+1)^2), arXiv:math/0607096 [math.NT], 2006.
J. Pintz, Landau's problems on primes
S. Ramanujan, A proof of Bertrand's postulate, J. Indian Math. Soc., 11 (1919), 181-182.
J. Sondow, Ramanujan Prime in MathWorld.
J. Sondow and E. W. Weisstein, Bertrand's Postulate in MathWorld.
Eric Weisstein's World of Mathematics, Legendre's Conjecture.

See A000720, A014085, A060715, A143223, A143225, A143226.

Cf. A104272, A143227. [Jonathan Sondow, Aug 03 2008]

with(numtheory): A143224:=n->`if`(pi((n+1)^2)-pi(n^2) = pi(2*n)-pi(n), n, NULL): seq(A143224(n), n=0..2000); # Wesley Ivan Hurt, Jul 25 2017

L={}; Do[If[PrimePi[(n+1)^2]-PrimePi[n^2] == PrimePi[2n]-PrimePi[n], L=Append[L,n]], {n,0,2000}]; L
(* Second program *)
With[{nn = 2000}, {0}~Join~Position[#, {0}][[All, 1]] &@ Map[Differences, Transpose@ {Differences@ Array[PrimePi[#^2] &, nn], Array[PrimePi[2 #] - PrimePi[#] &, nn - 1]}]] (* Michael De Vlieger, Jul 25 2017 *)

is(n) = primepi((n+1)^2)-primepi(n^2)==primepi(2*n)-primepi(n) \\ Felix Fröhlich, Jul 25 2017

A143223(a(n)) = 0.

A143225 Number of primes between n^2 and (n+1)^2, if equal to the number of primes between n and 2n.

Original entry on oeis.org

0, 3, 9, 9, 10, 10, 16, 20, 19, 21, 23, 23, 24, 25, 28, 31, 32, 36, 38, 56, 57, 59, 59, 62, 65, 71, 75, 84, 88, 88, 96, 102, 107, 115, 116, 119, 120, 126, 125, 129, 132, 132, 163, 168, 168, 182, 189, 189, 192, 197, 198, 213, 236

Offset: 1

Jonathan Sondow, Jul 31 2008

nonn

Legendre's conjecture (still open) says there is always a prime between n^2 and (n+1)^2. Bertrand's postulate (actually a theorem due to Chebyshev) says there is always a prime between n and 2n.

See the additional reference and link to Ramanujan's work mentioned in A143223. [Jonathan Sondow, Aug 03 2008]

			There are 3 primes between 9^2 and 10^2 and 3 primes between 9 and 2*9, so 3 is a member.

M. Aigner and C. M. Ziegler, Proofs from The Book, Chapter 2, Springer, NY, 2001.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 5th ed., Oxford Univ. Press, 1989, p. 19.

T. D. Noe, Table of n, a(n) for n=1..97 (no other n < 10^6)
T. Hashimoto, On a certain relation between Legendre's conjecture and Bertrand's postulate, arXiv:0807.3690 [math.GM], 2008.
M. Hassani, Counting primes in the interval (n^2,(n+1)^2), arXiv:math/0607096 [math.NT], 2006.
J. Pintz, Landau's problems on primes
S. Ramanujan, A proof of Bertrand's postulate, J. Indian Math. Soc., 11 (1919), 181-182.
J. Sondow, Ramanujan Prime in MathWorld
J. Sondow and E. W. Weisstein, Bertrand's Postulate in MathWorld
E. W. Weisstein, Legendre's Conjecture in MathWorld

See A000720, A014085, A060715, A143223, A143224, A143226.

Cf. A104272, A143227. [Jonathan Sondow, Aug 03 2008]

L={}; Do[If[PrimePi[(n+1)^2]-PrimePi[n^2] == PrimePi[2n]-PrimePi[n], L=Append[L,PrimePi[2n]-PrimePi[n]]], {n,0,2000}]; L

a(n) = A014085(A143224(n)) = A060715(A143224(n)) for n > 0.

A143226 Numbers n such that there are more primes between n and 2n than between n^2 and (n+1)^2.

Original entry on oeis.org

42, 55, 56, 58, 69, 77, 80, 119, 136, 137, 143, 145, 149, 156, 174, 177, 178, 188, 219, 225, 232, 247, 253, 254, 257, 261, 263, 297, 306, 310, 325, 327, 331, 335, 339, 341, 344, 356, 379, 395, 402, 410, 418, 421, 425, 433, 451, 485, 500

Offset: 1

Jonathan Sondow, Jul 31 2008

nonn

Legendre's conjecture (still open) says there is always a prime between n^2 and (n+1)^2. Bertrand's postulate (actually a theorem due to Chebyshev) says there is always a prime between n and 2n.
It appears that this sequence is finite; searching up to 10^5, the last n appears to be 48717. [T. D. Noe, Aug 01 2008]
If the sequence is finite, then, by Bertrand's postulate, Legendre's conjecture is true for sufficiently large n. - Jonathan Sondow, Aug 02 2008
No other n <= 10^6. The plot of A143223 shows that it is quite likely that there are no additional terms. - T. D. Noe, Aug 04 2008
See the additional reference and link to Ramanujan's work mentioned in A143223. - Jonathan Sondow, Aug 03 2008

			There are 10 primes between 42 and 2*42, but only 9 primes between 42^2 and 43^2, so 42 is a member.

M. Aigner and C. M. Ziegler, Proofs from The Book, Chapter 2, Springer, NY, 2001.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 5th ed., Oxford Univ. Press, 1989, p. 19.

T. D. Noe, Table of n, a(n) for n = 1..413
T. Hashimoto, On a certain relation between Legendre's conjecture and Bertrand's postulate, arXiv:0807.3690 [math.GM], 2008.
M. Hassani, Counting primes in the interval (n^2,(n+1)^2), arXiv:math/0607096 [math.NT], 2006.
J. Pintz, Landau's problems on primes
S. Ramanujan, A proof of Bertrand's postulate, J. Indian Math. Soc., 11 (1919), 181-182.
J. Sondow, Ramanujan Prime in MathWorld
J. Sondow and E. W. Weisstein, Bertrand's Postulate in MathWorld
E. W. Weisstein, Legendre's Conjecture in MathWorld

See A000720, A014085, A060715, A143223, A143224, A143225, A104272, A143227.

L={}; Do[If[PrimePi[(n+1)^2]-PrimePi[n^2] < PrimePi[2n]-PrimePi[n], L=Append[L,n]], {n,0,500}]; L

A143223(n) < 0.

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 *)

A168425 Large Associated Ramanujan Prime, p_i.

Original entry on oeis.org

3, 13, 19, 31, 43, 53, 61, 71, 73, 101, 103, 109, 131, 151, 157, 173, 181, 191, 229, 233, 239, 241, 251, 269, 271, 283, 311, 313, 349, 353, 373, 379, 409, 419, 421, 433, 439, 443, 463, 491, 499, 509, 571, 577, 593, 599, 601, 607, 613, 643, 647, 653, 659, 661

Offset: 1

John W. Nicholson, Nov 25 2009

nonn

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.
a(n) = nextprime(R_n) = nextprime(p_k), where nextprime(x) is the next prime larger than x.
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).
Proof of Corollary: See Wikipedia link.
The number of primes until the next Ramanujan prime, R_(n+1), can be found in A190874.
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
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

			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. D. Noe, Table of n, a(n) for n = 1..1000
S. Ramanujan, A proof of Bertrand's postulate, J. Indian Math. Soc., 11 (1919), 181-182.
Vladimir Shevelev, Ramanujan and Labos primes, their generalizations and classifications of primes, arXiv:0909.0715 [math.NT], 2009-2011.
Jonathan Sondow, Ramanujan primes and Bertrand's postulate, Amer. Math. Monthly 116 (2009) 630-635.
Jonathan Sondow, John W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, J. Integer Seq. 14 (2011) Article 11.6.2.
Jonathan Sondow, Ramanujan Prime in MathWorld
Anitha Srinivasan, An upper bound for Ramanujan primes, Integers, 19 (2014), #A19.
Wikipedia, Ramanujan Prime

Cf. A104272, A168421, A179196, A190874.

Cf. A202187, A202188, A234298.

nn = 100; 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]}
];
RamanujanPrimes = R + 1;
Prime[PrimePi[#]+1]& /@ RamanujanPrimes (* Jean-François Alcover, Nov 03 2018, after T. D. Noe in A104272 *)

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(snextprime(x+1),L);v} \\ Bill McEachen, Jun 24 2023 (incorporates code from A104272)

use ntheory ":all"; say next_prime(nth_ramanujan_prime($)) for 1..100; # _Dana Jacobsen, Dec 25 2015

a(n) = prime(primepi(A104272(n)) + 1).

a(n) = A151800(A104272(n)). - Michel Marcus, Jun 27 2023

A102820 Number of primes between 2prime(n) and 2prime(n+1), where prime(n) is the n-th prime.

Original entry on oeis.org

1, 1, 2, 2, 1, 2, 1, 2, 2, 2, 3, 1, 1, 1, 3, 3, 0, 2, 2, 0, 3, 1, 2, 4, 2, 0, 1, 0, 1, 6, 1, 3, 1, 3, 0, 3, 1, 1, 1, 3, 1, 3, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 3, 2, 2, 0, 1, 1, 1, 1, 3, 6, 2, 0, 1, 6, 1, 3, 0, 1, 1, 3, 2, 2, 1, 2, 1, 1, 2, 4, 1, 3, 1, 1, 2, 1, 2, 1, 0, 1, 4, 2, 1, 3, 0, 2, 5, 0, 5, 3, 3, 2, 1, 0, 2

Offset: 1

Ali A. Tanara (tanara(AT)khayam.ut.ac.ir), Feb 27 2005

Number of primes between successive even semiprimes. [Juri-Stepan Gerasimov, May 01 2010]
From Peter Munn, Jun 01 2023: (Start)
First differences of A020900.
A080192 lists prime(n) corresponding to the zero terms.
A104380(k) is prime(n) corresponding to the first occurrence of k as a term.
If a(n) is nonzero, A059786(n) is the smallest and A059788(n+1) the largest of the a(n) enumerated primes. In the tree of primes described in A290183, these primes label the child nodes of prime(n).
Conjecture: the asymptotic proportions of 0's, 1's, ... , k's, ... are 1/3, 2/9, ... , 2^k/3^(k+1), ... .
(End)

			a(15)=3 because there are 3 primes between the doubles of the 15th and 16th primes, that is between 2*47 and 2*53.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
V. Shevelev, On critical small intervals containing primes, arXiv:0908.2319 [math.NT], 2009.

Sequences with related analysis: A020900, A059786, A059788, A080192, A104380, A290183.
Cf. A104272, A080359. [Vladimir Shevelev, Aug 24 2009]
Cf. A100484, A010051.
Sequences with similar definitions: A104289, A217564.

a102820 n = a102820_list !! (n-1)
a102820_list =  map (sum . (map a010051)) $
   zipWith enumFromTo a100484_list (tail a100484_list)
-- Reinhard Zumkeller, Apr 29 2012

Table[PrimePi[2 Prime[n+1]]-PrimePi[2 Prime[n]], {n, 150}] (* Zak Seidov *)
Differences[PrimePi[2 Prime[Range[110]]]] (* Harvey P. Dale, Oct 29 2022 *)

a(n) = primepi(2*prime(n+1)) - primepi(2*prime(n)); \\ Michel Marcus, Sep 22 2017

a(n) = A020900(n+1) - A020900(n). - Peter Munn, Jun 01 2023

More terms from Zak Seidov, Feb 28 2005

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 *)

A174641 Smallest prime that begins a run of n consecutive primes that are not Ramanujan primes.

Original entry on oeis.org

3, 3, 3, 73, 191, 191, 509, 2539, 2539, 5279, 9901, 9901, 9901, 11593, 11593, 55343, 55343, 55343, 55343, 55343, 174929, 174929, 174929, 225977, 225977, 225977, 225977, 225977, 534889, 534889, 534889, 534889, 534889, 534889, 534889, 534889, 2492317, 2492317

Offset: 1

T. D. Noe, Nov 29 2010

nonn

The run of 10 consecutive non-Ramanujan primes was mentioned by Sondow.

Dana Jacobsen, Table of n, a(n) for n = 1..107 (first 67 terms from T. D. Noe)
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, arXiv:1105.2249 [math.NT], 2011.
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 (Ramanujan primes), A174635 (non-Ramanujan primes).
Cf. A174602 (runs of Ramanujan primes).
Cf. A202187, A202188.

nn=10000; t=Table[0, {nn}]; len=Prime[3*nn]; s=0; Do[If[PrimeQ[k], s++]; If[PrimeQ[k/2], s--]; If[s

use ntheory ":all";
my($k, $max, $start, $end, $inc, $p, $q, $r, $pi)
   = (0, 0, 0, 10, 1e9, 0, 2, [], prime_iterator(3));
while (1) {
  if (!@$r) {
    ($start, $end) = ($end+1, $end+$inc);
    $r = ramanujan_primes($start, $end);
  }
  ($p, $q, $k) = ($q, shift(@$r), 0);
  # $k = prime_count($p+1,$q-1);
  $k++ while $pi->() < $q;
  say ++$max," ",next_prime($p)   while $k > $max;
}
# Dana Jacobsen, Jul 14 2016

cp's OEIS Frontend

A194658 a(n) is the maximal prime, such that for all primes x<=a(n) the number of primes in (x/2,x) is less than n.

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A143223 (Number of primes between n^2 and (n+1)^2) - (number of primes between n and 2n).

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A143224 Numbers n such that (number of primes between n^2 and (n+1)^2) = (number of primes between n and 2n).

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PARI

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A143225 Number of primes between n^2 and (n+1)^2, if equal to the number of primes between n and 2n.

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A143226 Numbers n such that there are more primes between n and 2n than between n^2 and (n+1)^2.

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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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A168425 Large Associated Ramanujan Prime, p_i.

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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)).

A102820 Number of primes between 2prime(n) and 2prime(n+1), where prime(n) is the n-th prime.

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