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A192820 2-Ramanujan primes: the interval (x/2,x] has at least n Ramanujan primes for x >= a(n) but not for x = a(n) - 1.

%I A192820 #60 Oct 22 2022 15:02:23
%S A192820 11,41,59,97,149,151,227,229,233,239,263,307,367,373,401,409,569,571,
%T A192820 587,593,599,641,643,647,653,719,751,821,937,941,1009,1019,1021,1031,
%U A192820 1049,1051,1061,1063,1217,1367,1373,1423,1427,1439,1481,1487,1549,1553,1559
%N A192820 2-Ramanujan primes: the interval (x/2,x] has at least n Ramanujan primes for x >= a(n) but not for x = a(n) - 1.
%C A192820 It is conjectured that primepi(a(n)) <= 7*n for all n. - _T. D. Noe_, Aug 26 2011
%C A192820 Subsequence of the Ramanujan primes A104272, by the minimality of a(n). - _Jonathan Sondow_, Aug 21 2012
%C A192820 Paksoy (2012) denotes a(n) by R'_n and calls it "the n-th derived Ramanujan prime." He proves the bounds on R'_n below. - _Jonathan Sondow_, Oct 29 2012
%H A192820 T. D. Noe, <a href="/A192820/b192820.txt">Table of n, a(n) for n = 1..1000</a>
%H A192820 M. B. Paksoy, <a href="http://arxiv.org/abs/1210.6991">Derived Ramanujan primes: R'_n</a>, arXiv:1210.6991 [math.NT], 2012.
%F A192820 R(2n) <= a(n) < R(3n), where R(n) = the n-th Ramanujan prime (Paksoy 2012).
%F A192820 p(4n)  < a(n) < p(9n), where p(n) = the n-th prime (Paksoy 2012).
%F A192820 a(n) < p(8n) for n >= 5315 (Paksoy 2012).
%F A192820 R(2n) ~ a(n) ~ p(4n) as n -> oo (Paksoy 2012).
%Y A192820 Cf. A104272 (Ramanujan primes), A192821, A192822, A192823, A192824, A225907.
%K A192820 nonn
%O A192820 1,1
%A A192820 _T. D. Noe_, Jul 11 2011
%E A192820 Definition clarified by _Jonathan Sondow_, Aug 21 2012