%I #30 Mar 06 2022 05:34:56
%S -1,4,4,10,12,10,16,14,10,26,22,18,26,42,38,36,40,30,64,56,52,46,42,
%T 40,40,42,56,48,76,68,74,62,84,72,70,72,60,56,64,78,70,70,126,114,124,
%U 114,108,98,86,100,86,78,76,66
%N R(n) - prime(2n), where R(n) is the n-th Ramanujan prime and prime(n) is the n-th prime.
%C The sequence tends to decrease at runs of Ramanujan primes and at twin Ramanujan primes.
%C Is 4 the minimum value of a(n) for all n > 1? Is the sequence unbounded? What are its liminf and limsup? Is a(n)/n bounded?
%C Christian Axler has proved that the answers to the 1st, 2nd, and 4th questions are yes, and that liminf a(n) = limsup a(n) = infinity. - _Jonathan Sondow_, Feb 12 2014
%C a(n) > n, for 1 < n < 86853959 = limit. For limit, a(n) = 135595760, a(n) - n = 48741801. - _John W. Nicholson_, Dec 19 2013
%H John W. Nicholson, <a href="/A233739/b233739.txt">Table of n, a(n) for n = 1..10000</a>
%H Christian Axler, <a href="http://arxiv.org/abs/1401.7179">On generalized Ramanujan primes</a>, arXiv:1401.7179 [math.NT], 2014.
%H Jonathan Sondow, <a href="http://arxiv.org/abs/0907.5232">Ramanujan primes and Bertrand's postulate</a>, Amer. Math. Monthly, 116 (2009), 630-635; arXiv:0907.5232 [math.NT], 2009-2010.
%H 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>, arXiv:1105.2249 [math.NT], 2011; J. Integer Seq. 14 (2011) Article 11.6.2.
%F a(n) = A104272(n) - A000040(2n).
%F a(n) = 2*A233740(n) for n > 1.
%F a(n) >= 2 for n > 1 (see "Ramanujan primes and Bertrand's postulate").
%F a(n)/p(2n) = R(n)/p(2n) - 1 -> 0 as n -> infinity (see same link).
%e R(2) - prime(4) = 11 - 7 and R(3) - prime(6) = 17 - 13, so a(2) = a(3) = 4.
%t nn = 60; R = Table[0, {nn}]; s = 0;
%t Do[If[PrimeQ[k], s++]; If[PrimeQ[k/2], s--]; If[s < nn, R[[s + 1]] = k], {k, Prime[3 nn]}];
%t R = R + 1;
%t Table[R[[n]] - Prime[2 n], {n, 1, nn}] (* _Jean-François Alcover_, Nov 07 2018, using _T. D. Noe_'s code for R *)
%Y Cf. A000040, A104272, A233740. Records are A233741.
%K sign
%O 1,2
%A _Jonathan Sondow_, Dec 15 2013