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

1,1

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.

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.

Formula

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

a(n) = A190654(2n-1) = A190654(2n) - 2.

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A001359, A104272, A164371, A178128, A190654.

Cf. A181678 (number of twin Ramanujan prime pairs less than 10^n).

Sequence in context: A308895 A100723 A316589 * A307472 A209619 A031929

Adjacent sequences: A178124 A178125 A178126 * A178128 A178129 A178130

Keyword

nonn

Author

Jonathan Sondow, May 20 2010

Status

approved