A179196 Number of primes up to the n-th Ramanujan prime: A000720(A104272(n)).

1, 5, 7, 10, 13, 15, 17, 19, 20, 25, 26, 28, 31, 35, 36, 39, 41, 42, 49, 50, 51, 52, 53, 56, 57, 60, 63, 64, 69, 70, 73, 74, 79, 80, 81, 83, 84, 85, 89, 93, 94, 96, 104, 105, 107, 108, 109, 110, 111, 116, 117, 118, 119, 120, 123, 128, 129, 131, 133, 136, 140, 142, 143

Offset: 1

Author

John W. Nicholson, Jul 02 2010

Keywords

• nonn

Comments

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.

Prime index of A168421(n), that is A000720(A168421(n)), is equal to a(n) - n + 1. - John W. Nicholson, Sep 16 2015

Examples

			The 10th Ramanujan prime is 97, and pi(97) = 25, so a(10) = 25.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Christian Axler, On the number of primes up to the nth Ramanujan prime, arXiv:1711.04588 [math.NT], 2017.
• Christian Axler, On Ramanujan primes, Functiones et Approximatio Commentarii Mathematici (2019).
• S. Ramanujan, A proof of Bertrand's postulate, J. Indian Math. Soc., 11 (1919), 181-182.
• H. W. Shapiro, Iterates of arithmetic functions and a property of the sequence of primes, Pacific J. Math. Volume 3, Number 3 (1953), 647-655.
• J. Sondow, Ramanujan primes and Bertrand's postulate, Amer. Math. Monthly, 116 7(2009), 630-635.
• J. Sondow, Ramanujan primes and Bertrand's postulate, arXiv:0907.5232 [math.NT], 2009, 2010.
• J. Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, J. Integer Seq. 14 (2011) Article 11.6.2.
• Wikipedia, Ramanujan prime

Crossrefs

Cf. A168421, A168425.

Programs

Mathematica

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 *)

Perl

use ntheory ":all"; say prime_count(nth_ramanujan_prime($)) for 1..100; # _Dana Jacobsen, Dec 25 2015

Formula

a(n) = A000720(A104272(n)).

a(n) = rho(n) in the paper by Sondow, Nicholson, and Noe.

prime(a(n)) = R_n = A104272(n).

a(n) = A000720(A168421(n)) + n - 1. - John W. Nicholson, Sep 16 2015