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%I #56 Feb 29 2020 20:26:47
%S 1,5,7,10,13,15,17,19,20,25,26,28,31,35,36,39,41,42,49,50,51,52,53,56,
%T 57,60,63,64,69,70,73,74,79,80,81,83,84,85,89,93,94,96,104,105,107,
%U 108,109,110,111,116,117,118,119,120,123,128,129,131,133,136,140,142,143
%N Number of primes up to the n-th Ramanujan prime: A000720(A104272(n)).
%C a(n) = k = pi(p_k) = pi(R_n), where pi is the prime number counting function and R_n is the n-th Ramanujan prime. I.e., p_k, the k-th prime, is the n-th Ramanujan prime.
%C Prime index of A168421(n), that is A000720(A168421(n)), is equal to a(n) - n + 1. - _John W. Nicholson_, Sep 16 2015
%H Charles R Greathouse IV, <a href="/A179196/b179196.txt">Table of n, a(n) for n = 1..10000</a>
%H Christian Axler, <a href="https://arxiv.org/abs/1711.04588">On the number of primes up to the nth Ramanujan prime</a>, arXiv:1711.04588 [math.NT], 2017.
%H Christian Axler, <a href="https://doi.org/10.7169/facm/1824">On Ramanujan primes</a>, Functiones et Approximatio Commentarii Mathematici (2019).
%H S. Ramanujan, <a href="http://ramanujan.sirinudi.org/Volumes/published/ram24.html">A proof of Bertrand's postulate</a>, J. Indian Math. Soc., 11 (1919), 181-182.
%H H. W. Shapiro, <a href="http://projecteuclid.org/euclid.pjm/1103051336">Iterates of arithmetic functions and a property of the sequence of primes</a>, Pacific J. Math. Volume 3, Number 3 (1953), 647-655.
%H J. Sondow, <a href="http://www.jstor.org/stable/40391170">Ramanujan primes and Bertrand's postulate</a>, Amer. Math. Monthly, 116 7(2009), 630-635.
%H J. Sondow, <a href="http://arxiv.org/abs/0907.5232"> Ramanujan primes and Bertrand's postulate</a>, arXiv:0907.5232 [math.NT], 2009, 2010.
%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>, J. Integer Seq. 14 (2011) Article 11.6.2.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Ramanujan_prime">Ramanujan prime</a>
%F a(n) = A000720(A104272(n)).
%F a(n) = rho(n) in the paper by Sondow, Nicholson, and Noe.
%F prime(a(n)) = R_n = A104272(n).
%F a(n) = A000720(A168421(n)) + n - 1. - _John W. Nicholson_, Sep 16 2015
%e The 10th Ramanujan prime is 97, and pi(97) = 25, so a(10) = 25.
%t f[n_] := With[{s = Table[{k, PrimePi[k] - PrimePi[k/2]}, {k, Prime[3 n]}]}, Table[1 + First@ Last@ Select[s, Last@ # == i - 1 &], {i, n}]]; PrimePi@ f@ 63 (* _Michael De Vlieger_, Nov 14 2017, after _Jonathan Sondow_ at A104272 *)
%o (Perl) use ntheory ":all"; say prime_count(nth_ramanujan_prime($_)) for 1..100; # _Dana Jacobsen_, Dec 25 2015
%Y Cf. A168421, A168425.
%K nonn
%O 1,2
%A _John W. Nicholson_, Jul 02 2010
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