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.

Showing 1-4 of 4 results.

A001359 Lesser of twin primes.

Original entry on oeis.org

3, 5, 11, 17, 29, 41, 59, 71, 101, 107, 137, 149, 179, 191, 197, 227, 239, 269, 281, 311, 347, 419, 431, 461, 521, 569, 599, 617, 641, 659, 809, 821, 827, 857, 881, 1019, 1031, 1049, 1061, 1091, 1151, 1229, 1277, 1289, 1301, 1319, 1427, 1451, 1481, 1487, 1607
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Also, solutions to phi(n + 2) = sigma(n). - Conjectured by Jud McCranie, Jan 03 2001; proved by Reinhard Zumkeller, Dec 05 2002
The set of primes for which the weight as defined in A117078 is 3 gives this sequence except for the initial 3. - Rémi Eismann, Feb 15 2007
The set of lesser of twin primes larger than three is a proper subset of the set of primes of the form 3n - 1 (A003627). - Paul Muljadi, Jun 05 2008
It is conjectured that A113910(n+4) = a(n+2) for all n. - Creighton Dement, Jan 15 2009
I would like to conjecture that if f(x) is a series whose terms are x^n, where n represents the terms of sequence A001359, and if we inspect {f(x)}^5, the conjecture is that every term of the expansion, say a_n * x^n, where n is odd and at least equal to 15, has a_n >= 1. This is not true for {f(x)}^k, k = 1, 2, 3 or 4, but appears to be true for k >= 5. - Paul Bruckman (pbruckman(AT)hotmail.com), Feb 03 2009
A164292(a(n)) = 1; A010051(a(n) - 2) = 0 for n > 1. - Reinhard Zumkeller, Mar 29 2010
From Jonathan Sondow, May 22 2010: (Start)
About 15% of primes < 19000 are the lesser of twin primes. About 26% of Ramanujan primes A104272 < 19000 are the lesser of twin primes.
About 46% of primes < 19000 are Ramanujan primes. About 78% of the lesser of twin primes < 19000 are Ramanujan 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". (End)
Primes generated by sequence A040976. - Odimar Fabeny, Jul 12 2010
Primes of the form 2*n - 3 with 2*n - 1 prime n > 2. Primes of the form (n^2 - (n-2)^2)/2 - 1 with (n^2 - (n-2)^2)/2 + 1 prime so sum of two consecutive odd numbers/2 - 1. - Pierre CAMI, Jan 02 2012
Conjecture: For any integers n >= m > 0, there are infinitely many integers b > a(n) such that the number Sum_{k=m..n} a(k)*b^(n-k) (i.e., (a(m), ..., a(n)) in base b) is prime; moreover, when m = 1 there is such an integer b < (n+6)^2. - Zhi-Wei Sun, Mar 26 2013
Except for the initial 3, all terms are congruent to 5 mod 6. One consequence of this is that no term of this sequence appears in A030459. - Alonso del Arte, May 11 2013
Aside from the first term, all terms have digital root 2, 5, or 8. - J. W. Helkenberg, Jul 24 2013
The sequence provides all solutions to the generalized Winkler conjecture (A051451) aside from all multiples of 6. Specifically, these solutions start from n = 3 as a(n) - 3. This gives 8, 14, 26, 38, 56, ... An example from the conjecture is solution 38 from twin prime pairs (3, 5), (41, 43). - Bill McEachen, May 16 2014
Conjecture: a(n)^(1/n) is a strictly decreasing function of n. Namely a(n+1)^(1/(n+1)) < a(n)^(1/n) for all n. This conjecture is true for all a(n) <= 1121784847637957. - Jahangeer Kholdi and Farideh Firoozbakht, Nov 21 2014
a(n) are the only primes, p(j), such that (p(j+m) - p(j)) divides (p(j+m) + p(j)) for some m > 0, where p(j) = A000040(j). For all such cases m=1. It is easy to prove, for j > 1, the only common factor of (p(j+m) - p(j)) and (p(j+m) + p(j)) is 2, and there are no common factors if j = 1. Thus, p(j) and p(j+m) are twin primes. Also see A067829 which includes the prime 3. - Richard R. Forberg, Mar 25 2015
Primes prime(k) such that prime(k)! == 1 (mod prime(k+1)) with the exception of prime(991) = 7841 and other unknown primes prime(k) for which (prime(k)+1)*(prime(k)+2)*...*(prime(k+1)-2) == 1 (mod prime(k+1)) where prime(k+1) - prime(k) > 2. - Thomas Ordowski and Robert Israel, Jul 16 2016
For the twin prime criterion of Clement see the link. In Ribenboim, pp. 259-260 a more detailed proof is given. - Wolfdieter Lang, Oct 11 2017
Conjecture: Half of the twin prime pairs can be expressed as 8n + M where M > 8n and each value of M is a distinct composite integer with no more than two prime factors. For example, when n=1, M=21 as 8 + 21 = 29, the lesser of a twin prime pair. - Martin Michael Musatov, Dec 14 2017
For a discussion of bias in the distribution of twin primes, see my article on the Vixra web site. - Waldemar Puszkarz, May 08 2018
Since 2^p == 2 (mod p) (Fermat's little theorem), these are primes p such that 2^p == q (mod p), where q is the next prime after p. - Thomas Ordowski, Oct 29 2019, edited by M. F. Hasler, Nov 14 2019
The yet unproved "Twin Prime Conjecture" states that this sequence is infinite. - M. F. Hasler, Nov 14 2019
Lesser of the twin primes are the set of elements that occur in both A162566, A275697. Proof: A prime p will only have integer solutions to both (p+1)/g(p) and (p-1)/g(p) when p is the lesser of a twin prime, where g(p) is the gap between p and the next prime, because gcd(p+1,p-1) = 2. - Ryan Bresler, Feb 14 2021
From Lorenzo Sauras Altuzarra, Dec 21 2021: (Start)
J. A. Hervás Contreras observed the subsequence 11, 311, 18311, 1518311, 421518311... (see the links), which led me to conjecture the following statements.
I. If i is an integer greater than 2, then there exist positive integers j and k such that a(j) equals the concatenation of 3k and a(i).
II. If k is a positive integer, then there exist positive integers i and j such that a(j) equals the concatenation of 3k and a(i).
III. If i, j, and r are positive integers such that i > 2 and a(j) equals the concatenation of r and a(i), then 3 divides r. (End)

References

  • Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.
  • T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 6.
  • Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §3.2 Prime Numbers, p. 81.
  • Paulo Ribenboim, The New Book of Prime Number Records, Springer-Verlag NY 1996, pp. 259-260.
  • Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 192-197.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 111-112.

Links

Crossrefs

Subsequence of A003627.
Cf. A001223, A006512 (greater of twin primes), A014574, A001097, A077800, A002822, A040040, A054735, A067829, A082496, A088328, A117078, A117563, A074822, A071538, A007508, A146214, A350246, A350247.
Cf. A104272 (Ramanujan primes), A178127 (lesser of twin Ramanujan primes), A178128 (lesser of twin primes if it is a Ramanujan prime).
Cf. A010051, A000040.

Programs

  • Haskell
    a001359 n = a001359_list !! (n-1)
a001359_list = filter ((== 1) . a010051' . (+ 2)) a000040_list
-- Reinhard Zumkeller, Feb 10 2015
  • Magma
    [n: n in PrimesUpTo(1610) | IsPrime(n+2)];  // Bruno Berselli, Feb 28 2011
  • Maple
    select(k->isprime(k+2),select(isprime,[$1..1616])); # Peter Luschny, Jul 21 2009
A001359 := proc(n)
   option remember;
   if n = 1
      then 3;
   else
      p := nextprime(procname(n-1)) ;
      while not isprime(p+2) do
         p := nextprime(p) ;
      end do:
      p ;
   end if;
end proc: # R. J. Mathar, Sep 03 2011
  • Mathematica
    Select[Prime[Range[253]], PrimeQ[# + 2] &] (* Robert G. Wilson v, Jun 09 2005 *)
a[n_] := a[n] = (p = NextPrime[a[n - 1]]; While[!PrimeQ[p + 2], p = NextPrime[p]]; p); a[1] = 3; Table[a[n], {n, 51}]  (* Jean-François Alcover, Dec 13 2011, after R. J. Mathar *)
nextLesserTwinPrime[p_Integer] := Block[{q = p + 2}, While[NextPrime@ q - q > 2, q = NextPrime@ q]; q]; NestList[nextLesserTwinPrime@# &, 3, 50] (* Robert G. Wilson v, May 20 2014 *)
Select[Partition[Prime[Range[300]],2,1],#[[2]]-#[[1]]==2&][[All,1]] (* Harvey P. Dale, Jan 04 2021 *)
q = Drop[Prepend[p = Prime[Range[100]], 2], -1];
Flatten[q[[#]] & /@ Position[p - q, 2]] (* Horst H. Manninger, Mar 28 2021 *)
  • PARI
    A001359(n,p=3) = { while( p+2 < (p=nextprime( p+1 )) || n-->0,); p-2}
/* The following gives a reasonably good estimate for any value of n from 1 to infinity; compare to A146214. */
A001359est(n) = solve( x=1,5*n^2/log(n+1), 1.320323631693739*intnum(t=2.02,x+1/x,1/log(t)^2)-log(x) +.5 - n)
/* The constant is A114907; the expression in front of +.5 is an estimate for A071538(x) */ \\  M. F. Hasler, Dec 10 2008
  • Python
    from sympy import primerange, isprime
print([n for n in primerange(1, 2001) if isprime(n + 2)]) # Indranil Ghosh, Jul 20 2017

Formula

a(n) = A077800(2n-1).
A001359 = { n | A071538(n-1) = A071538(n)-1 }; A071538(A001359(n)) = n. - M. F. Hasler, Dec 10 2008
A001359 = { prime(n) : A069830(n) = A087454(n) }. - Juri-Stepan Gerasimov, Aug 23 2011
a(n) = prime(A029707(n)). - R. J. Mathar, Feb 19 2017

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

A178127 Lesser of twin Ramanujan primes.

Original entry on oeis.org

149, 179, 227, 239, 347, 431, 569, 599, 641, 821, 1019, 1049, 1061, 1427, 1487, 1607, 1787, 1997, 2081, 2129, 2237, 2267, 2657, 2687, 2711, 2789, 2999, 3167, 3257, 3299, 3359, 3527, 3539, 3581, 3671, 3917, 4091, 4127, 4229, 4241, 4337, 4547, 4637, 4649
Offset: 1

Views

Author

Jonathan Sondow, May 20 2010

Keywords

Comments

By definition, a number p is a member if p and p+2 are Ramanujan primes A104272.
Conjecture: For all n > 570, more than 1/4 of the twin prime pairs < n are both Ramanujan primes.
Motivation for the conjecture is in "Ramanujan primes and Bertrand's postulate" Section 7.
Subsequence of A178128.
See A001359 and A104272 for additional comments, links, and references.

Examples

			a(1) = 149 because (149,151) is the 1st pair of twin primes both of which are Ramanujan primes.
11 is not a member even though 11 and 13 are twin primes and 11 is a Ramanujan prime, because 13 is not also a Ramanujan prime.

Links

Crossrefs

Cf. A001359, A104272, A164371, A178128, A190654.
Cf. A181678 (number of twin Ramanujan prime pairs less than 10^n).

Programs

  • Maple
    n := 1:
for i from 1 do
    if A104272(i+1) = A104272(i)+2 then
        printf("%d %d\n",n,A104272(i)) ;
        n := n+1 ;
    end if;
end do: # produces b-file, R. J. Mathar, Sep 21 2017
  • Mathematica
    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]}];
A104272 = R + 1;
twins1 = Position[A104272 // Differences, 2] // Flatten;
A104272[[twins1]] (* Jean-François Alcover, Oct 28 2018, after T. D. Noe in A104272 *)
  • Perl
    use ntheory ":all"; my $r = ramanujan_primes(1e5); my @rt = @$r[grep { $r->[$+1]-$r->[$]==2 } 0..$#$r-1]; say "@rt"; # Dana Jacobsen, Sep 06 2015

Formula

{A104272(n): A104272(n+1) = A104272(n)+2}.
a(n) = A190654(2n-1) = A190654(2n) - 2.

A173081 Number of twin prime pairs < 10^n that contain at least one Ramanujan prime (A104272).

Original entry on oeis.org

0, 6, 28, 167, 964, 6305, 45082, 335919, 2605867, 20841010, 170395131
Offset: 1

Views

Author

T. D. Noe, Nov 22 2010

Keywords

Comments

It appears that this gives the number of Ramanujan primes < 10^n that are the lesser prime in a twin prime pair. Equivalently, this sequence also gives the number of Ramanujan primes p with p+2 also prime less than 10^n.
It appears that no upper twin prime is a Ramanujan prime without the corresponding lower twin prime also being a Ramanujan prime.
This is proved in Section 4 of "Ramanujan Primes: Bounds, Runs, Twins, and Gaps".

Links

Crossrefs

Cf. A178128 (Ramanujan primes p such that p+2 is prime), A007508 (number of twin primes pairs < 10^n), A181678 (number of twin Ramanujan primes pairs < 10^n).

Programs

  • Mathematica
    nn=50000; t=Table[0,{nn}]; s=0; Do[If[PrimeQ[k], s++]; If[PrimeQ[k/2], s--]; If[s

Extensions

a(10)-a(11) from Dana Jacobsen, Apr 29 2015
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.