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A233822 a(n) = 2*R(n) - R(n+1), where R(n) is the n-th Ramanujan prime.

%I A233822 #15 Dec 06 2018 10:12:36
%S A233822 -7,5,5,17,35,35,51,63,45,93,95,87,105,147,135,155,177,135,225,225,
%T A233822 227,237,219,257,257,255,303,275,345,331,361,345,393,399,407,429,427,
%U A233822 417,435,483,479,437,567,555,581,587,597,595,573,639,639,641,647
%N A233822 a(n) = 2*R(n) - R(n+1), where R(n) is the n-th Ramanujan prime.
%C A233822 a(n) = 2*A104272(n) - A104272(n+1).
%C A233822 Paksoy proved that a(n) > 0 for n > 1.
%C A233822 Paksoy's theorem is the analog for Ramanujan primes of Chebychev's theorem (Bertrand's postulate) that 2*prime(n) - prime(n+1) > 0 for n > 0 (see A062234).
%H A233822 Dana Jacobsen, <a href="/A233822/b233822.txt">Table of n, a(n) for n = 1..10000</a>
%H A233822 Baris Paksoy, <a href="http://arxiv.org/abs/1210.6991">Derived Ramanujan primes: R'_n</a>, arXiv:1210.6991 [math.NT], 2012.
%e A233822 The only negative term is a(1) = 2*R(1) - R(2) = 2*2 - 11 = -7.
%t A233822 nn = 100;
%t A233822 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]}]; R = R + 1;
%t A233822 Most[2 R - RotateLeft[R]] (* _Jean-François Alcover_, Dec 06 2018, after _T. D. Noe_ in A104272 *)
%o A233822 (Perl) use ntheory ":all"; say 2*nth_ramanujan_prime($_)-nth_ramanujan_prime($_+1) for 1..10 # _Dana Jacobsen_, Sep 02 2017
%Y A233822 Cf. A062234, A104272, A225907.
%K A233822 sign
%O A233822 1,1
%A A233822 _Jonathan Sondow_, Dec 16 2013