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%I #48 Oct 27 2019 12:00:08
%S 2,3,5,5,5,11,3,7,3,9,5,11,7,9,7,11,15,13,27,25,21,15,13,11,5,17,7,3,
%T 11,9,15,9,21,13,3,15,13,7,5,15,11,11,17,15,27,21,15,13,7,21,19,15,9,
%U 3,17,15,7,7,7,9,9,17,15,11,9,5,5,21,17,11,7,15,9
%N Size of the range of the Ramanujan Prime Corollary, 2*A168421(n) - A104272(n).
%C All but the first term is odd because A104272 has only one even term, 2. Because of all primes > 2 are odd, 1 can be subtracted from each term.
%C If this sequence has an infinite number of terms in which a(n) = 3, then the twin prime conjecture can be proved.
%C R_n is the sequence A104272(n) and k = pi(R_n)= A000720(R_n) with i>k.
%C By comparing the fractions we can see that (p_(i+1)-p_i)/(2*sqrt(p_i)) and a(n)/(2*sqrt(p_k)) are < 1 for all n > 0, in fact a(n)/(1.8*sqrt(p_k)) < 1 for all n > 0. When taking into account numbers in A182873(n) and A190874(n) to sqrt(R_n) we see that A182873(n)/(A190874(n)*sqrt(R_n)) < 1 for all n > 1.
%H T. D. Noe, <a href="/A165959/b165959.txt">Table of n, a(n) for n = 1..10000</a>
%H J. Sondow, <a href="http://arxiv.org/abs/0907.5232"> Ramanujan primes and Bertrand's postulate</a>, arXiv:0907.5232 [math.NT], 2009-201; Amer. Math. Monthly 116 (2009) 630-635.
%H J. 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.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Ramanujan_prime">Ramanujan Prime</a>
%H Marek Wolf, <a href="http://arxiv.org/abs/1010.3945">A Note on the Andrica Conjecture</a>, arXiv:1010.3945 [math.NT], 2010.
%F a(n) = 2*A168421(n) - A104272(n).
%e A168421(19) = 127, A104272(19) = 227; so a(19) = 2*A168421(19) - A104272(19) = 254 - 227 = 27. Note: for n = 20, 21, 22, 23 A168421(n) = 127. Because A168421 remains the same for these n and A104272 increases, the size of the range for a(n) for these n decreases. Note: a(18) = 2*97 - 181 = 194 - 181 = 13. This is nearly half a(19). The actual gap betweens A104272(19) and the next prime, 229, is 2.
%t nn = 100; 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 A104272 = R + 1; t = Table[0, {nn}];
%t Do[m = PrimePi[2 n] - PrimePi[n]; If[0 < m <= nn, t[[m]] = n], {n, 15 nn}];
%t A168421 = NextPrime[Join[{1}, t]] // Most;
%t A165979 = 2 A168421 - A104272 (* _Jean-François Alcover_, Nov 07 2018, after _T. D. Noe_ in A104272 *)
%Y Cf. A168421, A104272, A182873, A190874.
%K nonn
%O 1,1
%A _John W. Nicholson_, Sep 12 2011
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