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A178127 Lesser of twin Ramanujan primes.

%I A178127 #36 Mar 23 2022 04:29:46
%S A178127 149,179,227,239,347,431,569,599,641,821,1019,1049,1061,1427,1487,
%T A178127 1607,1787,1997,2081,2129,2237,2267,2657,2687,2711,2789,2999,3167,
%U A178127 3257,3299,3359,3527,3539,3581,3671,3917,4091,4127,4229,4241,4337,4547,4637,4649
%N A178127 Lesser of twin Ramanujan primes.
%C A178127 By definition, a number p is a member if p and p+2 are Ramanujan primes A104272.
%C A178127 Conjecture: For all n > 570, more than 1/4 of the twin prime pairs < n are both Ramanujan primes.
%C A178127 Motivation for the conjecture is in "Ramanujan primes and Bertrand's postulate" Section 7.
%C A178127 Subsequence of A178128.
%C A178127 See A001359 and A104272 for additional comments, links, and references.
%H A178127 Amiram Eldar, <a href="/A178127/b178127.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..674 from R. J. Mathar)
%H A178127 B. Ghusayni, <a href="http://ijmcs.future-in-tech.net/7.2/R-Ghusayni-subsets.pdf">Subsets of prime numbers</a>, Int. J. Math. Comp. Sci. 7 (2) 2012
%H A178127 Jonathan Sondow, <a href="http://arxiv.org/abs/0907.5232"> Ramanujan primes and Bertrand's postulate</a>, arXiv:0907.5232 [math.NT], 2009-2010.
%H A178127 Jonathan Sondow, <a href="http://www.jstor.org/stable/40391170">Ramanujan primes and Bertrand's postulate</a>, Amer. Math. Monthly, 116 (2009), 630-635.
%H A178127 Jonathan Sondow, J. W. Nicholson, and T. D. Noe, <a href="http://arxiv.org/abs/1105.2249"> Ramanujan Primes: Bounds, Runs, Twins, and Gaps</a>, J. Integer Seq. 14 (2011) Article 11.6.2.
%F A178127 {A104272(n): A104272(n+1) = A104272(n)+2}.
%F A178127 a(n) = A190654(2n-1) = A190654(2n) - 2.
%e A178127 a(1) = 149 because (149,151) is the 1st pair of twin primes both of which are Ramanujan primes.
%e A178127 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.
%p A178127 n := 1:
%p A178127 for i from 1 do
%p A178127     if A104272(i+1) = A104272(i)+2 then
%p A178127         printf("%d %d\n",n,A104272(i)) ;
%p A178127         n := n+1 ;
%p A178127     end if;
%p A178127 end do: # produces b-file, _R. J. Mathar_, Sep 21 2017
%t A178127 nn = 1000; R = Table[0, {nn}]; s = 0;
%t A178127 Do[If[PrimeQ[k], s++]; If[PrimeQ[k/2], s--]; If[s < nn, R[[s + 1]] = k], {k, Prime[3*nn]}];
%t A178127 A104272 = R + 1;
%t A178127 twins1 = Position[A104272 // Differences, 2] // Flatten;
%t A178127 A104272[[twins1]] (* _Jean-François Alcover_, Oct 28 2018, after _T. D. Noe_ in A104272 *)
%o A178127 (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
%Y A178127 Cf. A001359, A104272, A164371, A178128, A190654.
%Y A178127 Cf. A181678 (number of twin Ramanujan prime pairs less than 10^n).
%K A178127 nonn
%O A178127 1,1
%A A178127 _Jonathan Sondow_, May 20 2010