cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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.

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

Views

Author

Jonathan Sondow, Feb 27 2005

Keywords

Comments

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

Examples

			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]

References

  • 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.

Links

Crossrefs

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.

Programs

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

Formula

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

A084140 a(n) is the smallest number j such that if x >= j there are at least n primes between x and 2x exclusively.

Original entry on oeis.org

2, 6, 9, 15, 21, 24, 30, 34, 36, 49, 51, 54, 64, 75, 76, 84, 90, 91, 114, 115, 117, 120, 121, 132, 135, 141, 154, 156, 174, 175, 184, 187, 201, 205, 210, 216, 217, 220, 231, 244, 246, 252, 285, 286, 294, 297, 300, 301, 304, 321, 322, 324, 327, 330, 339, 360, 364
Offset: 1

Views

Author

Harry J. Smith, May 15 2003

Keywords

Comments

For all m >= a(n) there are at least n primes between m and 2m exclusively. This calculation relies on the fact that pi(2m) - pi(m) > m/(3*log(m)) for m >= 5. This is one more than the terms of A084139 with offset changed from 0 to 1.
For n > 5889, pi(2n) - pi(n) > f(2, 2n) - f(3, n) where f(k, x) = x/log x * (1 + 1/log x + k/(log x)^2). This may be useful for checking larger terms. The constant 3 can be improved at the cost of an increase in the constant 5889. - Charles R Greathouse IV, May 02 2012
A168421(n) = nextprime(a(n)), where nextprime(x) is the next prime >= x. - John W. Nicholson, Dec 21 2012
a(1) = ceiling((A104272(1)+1)/2) modifies the only even prime, 2; which has been stated, in Formula, as a(1) = A104272(1); for all others, a(n) = (A104272(n)+1)/2 = ceiling ((A104272(n)+1)/2). - John W. Nicholson, Dec 24 2012
Srinivasan's Lemma (2014): previousprime(a(n)) = p_(k-n) < (p_k)/2, where the n-th Ramanujan Prime R_n is the k-th prime p_k, and with 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. - Copied and adapted from a comment by Jonathan Sondow in A168421 by John W. Nicholson, Feb 17 2015

Examples

			a(11)=51 since there are at least 11 primes between m and 2m for all m >= 51 and this is not true for any m < 51. Although a(100)=720 is not listed, for all m >= 720, there are at least 100 primes between m and 2m.

References

  • Paulo Ribenboim, The Little Book of Big Primes, Springer-Verlag, 1991, p. 140.
  • Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag, 2004, p. 181.

Links

Crossrefs

Cf. A108954, A060715, A060756, A084138, A084139, A084141, A084142, A168421, A168425.

Programs

Formula

a(1) = A104272(1); for n >= 2, a(n) = (A104272(n)+1)/2. - Vladimir Shevelev, Dec 07 2012
a(n) = ceiling((A104272(n)+1)/2) for n >= 1. - John W. Nicholson, Dec 24 2012

A168421 Small Associated Ramanujan Prime, p_(i-n).

Original entry on oeis.org

2, 7, 11, 17, 23, 29, 31, 37, 37, 53, 53, 59, 67, 79, 79, 89, 97, 97, 127, 127, 127, 127, 127, 137, 137, 149, 157, 157, 179, 179, 191, 191, 211, 211, 211, 223, 223, 223, 233, 251, 251, 257, 293, 293, 307, 307, 307, 307, 307, 331, 331, 331
Offset: 1

Views

Author

John W. Nicholson, Nov 25 2009

Keywords

Comments

a(n) is the smallest prime p_(k+1-n) on the left side of the Ramanujan Prime Corollary, 2*p_(i-n) > p_i for i > k, where the n-th Ramanujan Prime R_n is the k-th prime p_k. [Comment clarified and shortened by Jonathan Sondow, Dec 20 2013]
Smallest prime number, a(n), such that if x >= a(n), then there are at least n primes between x and 2x exclusively.
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.
Not the same as A124136.
A084140(n) is the smallest integer where ceiling ((A104272(n)+1)/2), a(n) is the next prime after A084140(n). - John W. Nicholson, Oct 09 2013
If a(n) is in A005382(k) then A005383(k) is a twin prime with the Ramanujan prime, A104272(n) = A005383(k) - 2, and A005383(k) = A168425(n). If this sequence has an infinite number of terms in A005382, then the twin prime conjecture can be proved. - John W. Nicholson, Dec 05 2013
Except for A000101(1)=3 and A000101(2)=5, A000101(k) = a(n). Because of the large size of a gap, there are many repeats of the prime number in this sequence. - John W. Nicholson, Dec 10 2013
For some n and k, we see that a(n) = A104272(k) as to form a chain of primes similar to a Cunningham chain. For example (and the first example), a(2) = 7, links A104272(2) = 11 = a(3), links A104272(3) = 17 = a(4), links A104272(4) = 29 = a(6), links A104272(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, Dec 14 2013
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 Small Associated Ramanujan Prime, a(n) is not a Ramanujan prime for many values of n. - Jonathan Sondow, May 10 2014
Prime index of a(n), pi(a(n)) = i-n, is equal to A179196(n) - n + 1. - John W. Nicholson, Sep 15 2015
All maximal prime pairs in A002386 and A000101 are bounded by, for a particular n and i, the prime A104272(n) and twice a prime in A000040() following a(n). This means the gap between maximal prime pair cannot be more than twice the prior maximal prime gap. - John W. Nicholson, Feb 07 2019

Examples

			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, 53 is the small associated Ramanujan prime.

Links

Crossrefs

Cf. A000101, A005382, A005383, A084139, A084140.
Cf. A104272, A124136, A168425, A179196, A190874.
Cf. A165959 (range size), A230147 (records).

Programs

Formula

a(n) = prime(primepi(A104272(n)) + 1 - n).
a(n) = nextprime(A084139(n+1)), where nextprime(x) is the next prime > x. Note: some A084139(n) may be prime, therefore nextprime(x) not equal to x. - John W. Nicholson, Oct 11 2013
a(n) = nextprime(A084140(n)). - John W. Nicholson, Oct 11 2013

Extensions

Extended by T. D. Noe, Nov 22 2010

A084141 a(n) is the index in sequence A084138 when the value of that sequence is one (1), i.e., there is exactly one case where there are exactly a(n) primes between m and 2m, exclusively, for m > 0.

Original entry on oeis.org

0, 22, 36, 47, 79, 98, 114, 134, 173, 178, 184, 210, 218, 221, 245, 254, 262, 284, 297, 305, 327, 333, 373, 387, 396, 426, 459, 466, 470, 484, 530, 544, 563, 567, 575, 587, 616, 650, 694, 700, 706, 708, 737, 776, 859, 881, 885, 898, 926, 939, 974, 993, 1002
Offset: 0

Views

Author

Harry J. Smith, May 15 2003

Keywords

Comments

This calculation relies on the fact that Pi(2*m) - Pi(m) > m/(3*log(m)) for m >= 5. Conjecture: There are an infinite number of terms in this sequence.

Examples

			a(3)=47 because the 3rd one in sequence A084138 is its item 47. There is exactly one case where there are exactly 47 primes between m and 2m.

References

  • P. Ribenboim, The Little Book of Big Primes. Springer-Verlag, 1991, p. 140.

Links

Crossrefs

Cf. A060715, A060756, A084138, A084139, A084140, A084142.

A084138 a(n) is the number of times n is in sequence A060715, i.e., there are exactly a(n) cases where there are exactly n primes between m and 2m, exclusively, for m > 0.

Original entry on oeis.org

1, 3, 4, 4, 7, 3, 5, 6, 2, 9, 6, 2, 5, 10, 7, 8, 5, 3, 9, 10, 6, 4, 1, 8, 6, 5, 5, 9, 11, 10, 6, 6, 10, 8, 5, 6, 1, 3, 8, 9, 9, 5, 18, 16, 5, 7, 3, 1, 3, 12, 5, 3, 3, 3, 9, 8, 16, 7, 5, 8, 15, 10, 4, 2, 8, 7, 10, 13, 17, 5, 8, 7, 9, 10, 3, 5, 3, 6, 6, 1, 6, 8, 3, 3, 10, 15, 14, 16, 7, 10, 14, 5, 5, 3, 8
Offset: 0

Views

Author

Harry J. Smith, May 15 2003

Keywords

Comments

This calculation relies on the fact that Pi(2*m) - Pi(m) > m/(3*log(m)) for m >= 5. It can be shown that a(n) is never zero, i.e., every nonnegative integer is in sequence A060715.

Examples

			a(22)=1 because there are 22 primes between 120 and 240 (namely, prime numbers p(31)=127 through p(52)=239), and in no other case are there exactly 22 primes between m and 2m exclusively.

References

  • P. Ribenboim, The Little Book of Big Primes. Springer-Verlag, 1991, p. 140.

Links

Crossrefs

Cf. A060715, A060756, A084139, A084140, A084141, A084142.

A084142 Positive numbers k such that the number of primes between k and 2*k is different from the number of primes between m and 2*m for every number m != k.

Original entry on oeis.org

1, 120, 216, 300, 531, 714, 804, 999, 1344, 1356, 1395, 1680, 1764, 1770, 1959, 2046, 2121, 2325, 2484, 2511, 2760, 2826, 3150, 3285, 3396, 3744, 4044, 4116, 4146, 4314, 4710, 4839, 5046, 5070, 5136, 5250, 5586, 5970, 6411, 6459, 6501, 6504, 6846, 7275
Offset: 1

Views

Author

Harry J. Smith, May 15 2003

Keywords

Comments

The number of primes between k and 2*k is unique because no other number m > 0 has the same of primes between m and 2m, exclusively. k is the value of A060756(j) or A084139(j) when A084138(j) = 1. Question: Is this sequence infinitely long?
If k > 1 is a term then A060715(k-1) < A060715(k) < A060715(k+1). Consequently, (2*k-1, 2*k+1) is a twin prime pair, so 3|k. Additionally, it can be shown that k-1..k+3 are all composite numbers. Moreover, if k is even, then k-4..k+4 are all composite numbers. - Jon E. Schoenfield, Oct 08 2023

Examples

			120 is a term because there are 22 primes between 120 and 240 and no other number m > 0 has 22 primes between m and 2*m.

References

  • P. Ribenboim, The Little Book of Big Primes. Springer-Verlag, 1991, p. 140.

Links

Crossrefs

Cf. A060715, A060756, A084138, A084139, A084140, A084141.

Extensions

Name edited by Jon E. Schoenfield, Oct 08 2023

A124136 The list of primes p such that the number of primes in the open interval (p,2*p) is larger than the number of primes in the open interval (q,2*q) for all q

Original entry on oeis.org

2, 7, 11, 17, 23, 29, 31, 37, 53, 59, 71, 79, 89, 97, 101, 127, 137, 149, 157, 179, 191, 211, 223, 233, 251, 257, 263, 293, 307, 311, 331, 347, 367, 373, 379, 389, 409, 419, 431, 443, 457, 479, 487, 499, 521, 541, 547, 557, 563, 587, 599, 613, 617, 631, 641
Offset: 1

Views

Author

Jani Melik, Nov 30 2006

Keywords

Comments

Sequence A060715(n) lists the number of primes in the open interval (n,2*n).
If we extract its sublist for n a prime, the number of primes in the open interval (p,2*p), we have A070046(m) = 1, 1, 1, 2, 3, 3, 4, 4, 5, 6, 7, 9, 9, 9, 9, 11, 13, 12, 13, 14, 13, 15, 15, 16... for the primes p=2, 3, 5, 7, 11, 13, 17, 19 etc.
This sequence lists the primes p = prime(m) that set a new record in A070046(m).
Alternative definition: primes p defined by positions of records in A063124.

Examples

			a(1)=prime(1)=2 with 1 prime in the interval (2,4). a(2) is neither 3 (with 1 prime in the interval (3,6)), nor 5 (with 1 prime in the interval (5,10)), but a(2)=7 with 2 primes in the interval (7,14).
The primes 41, 43 and 47 are not in the list because the intervals (41,82), (43,86) and (47,94) contain 9 primes, but the interval (37,74) with the smaller prime p=37 already contained 9 primes.
The prime 53 is in the list because the interval (53,106) contains 11 primes and the intervals (q,2*q) for primes q =2,3, 5, ..,47 contained 9 or less primes.

Crossrefs

Cf. A060756, A060715, A084139.

Programs

  • Maple
    ts_c:=proc(n) local i,j,st_p,max_stp,ans; ans:= [ ]: st_p:=0: max_stp:=0: for i from 2 to n do for j from i+1 to 2*i-1 do if (isprime(j) = 'true') then st_p:=st_p+1: fi od: if (st_p > max_stp and isprime(i) = 'true') then max_stp := st_p: ans:=[ op(ans),i ]: fi; st_p:=0: od: RETURN(ans) end: ts_c(1200);

Extensions

Definition recovered from the Maple program. - R. J. Mathar, May 21 2025
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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