%I #8 Nov 05 2013 11:15:12
%S 4,11,19,28,38,48,58,69,80,91,102,114,126,138,150,162,174,187,200,212,
%T 225,238,251,265,278,291,305,318,332,345,359,373,387,401,415,429,443,
%U 458,472,486,501,515,530,544,559,573,588,603,618,632,647,662,677,692
%N a(n) = Round(kn * (log(kn)+1)), with k = 2.216 as an approximation of R_n = n-th Ramanujan prime A104272(n) and Abs(a(n)-R_n) < 2 * Sqrt(a(n)) for n in [1..1000].
%C a(n) approximates the {kn}-th prime number, which in turn approximates the n-th Ramanujan prime, and k = 2.216 is nearly optimal for n in [1..1000] since a(n) - 2*sqrt(a(n)) < R_n < a(n) + 2*sqrt(a(n)) in that range. Since R_n ~ Prime(2n) ~ 2n * (log(2n)+1) ~ 2n * log(2n), whereas A162996(n) ~ Prime(kn) ~ kn * (log(kn)+1) ~ kn * log(kn), giving A162996(n) / R_n ~ k/2 = 2.216/2 = 1.108 which implies an asymptotic overestimate of about 10% (a better approximation would need k to depend on n and be asymptotic to 2).
%C R_n is the smallest number such that if x >= R_n, then pi(x) - pi(x/2) >= n, where pi(x) is the number of primes <= x.
%H Daniel Forgues, <a href="/A162996/b162996.txt">Table of n, a(n) for n=1..1000</a>
%Y Cf. A163160 (Round(kn * (log(kn)+1)) - R_n, where k = 2.216 and R_n = n-th Ramanujan prime).
%Y Cf. A104272 (Ramanujan primes).
%K nonn
%O 1,1
%A _Daniel Forgues_, Jul 21 2009, Jul 29 2009