A178127 Lesser of twin Ramanujan primes.

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

Author

Jonathan Sondow, May 20 2010

Keywords

• nonn

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

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..674 from R. J. Mathar)
• B. Ghusayni, Subsets of prime numbers, Int. J. Math. Comp. Sci. 7 (2) 2012
• 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.
• Jonathan Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, J. Integer Seq. 14 (2011) Article 11.6.2.

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.