A190413 primepi(R_{n*m}) <= n*primepi(R_m) for m >= a(n), where R_k is the k-th Ramanujan prime (A104272).

1, 1245, 189, 189, 85, 85, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset

1,2

Comments

This is Conjecture 1 in the paper by Sondow, Nicholson, and Noe. The conjecture has been verified for n <= 20 and Ramanujan primes less than 10^9.

A restatement is rho(n*m) <= n*rho(m) for m >= a(n), where rho = A179196.

The conjecture has been proven for n > 10^300 by Shichun Yang and Alain Togbé. - Jonathan Sondow, Jan 21 2016

The conjecture has been proven for n > 38 and m > 9 by Christian Axler. Complete exception list can be found in remark of paper. - John W. Nicholson, Aug 04 2019

Links

Table of n, a(n) for n=1..79.

Christian Axler, On the number of primes up to the n-th Ramanujan prime, arXiv:1711.04588 [math.NT], 2017.

Jonathan Sondow, Ramanujan primes and Bertrand's postulate, Amer. Math. Monthly, Vol. 116, No. 7 (2009), 630-635; arXiv preprint, arXiv:0907.5232 [math.NT], 2009-2010.

Jonathan Sondow, John W. Nicholson, and Tony D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, J. Integer Seq., Vol. 14 (2011), Article 11.6.2; arXiv preprint, arXiv:1105.2249 [math.NT], 2011.

Shichun Yang and Alain Togbé, On the estimates of the upper and lower bounds of Ramanujan primes, Ramanujan J., Vol. 40 (2016), pp. 245-255.

Formula

For all n >= 20, a(n) = 2.

Crossrefs

Cf. A007395, A104272, A179196, A190414.

Sequence in context: A181073 A023065 A375801 * A252675 A066696 A195006

Adjacent sequences: A190410 A190411 A190412 * A190414 A190415 A190416

Keyword

nonn,easy

Author

T. D. Noe, May 11 2011

Status

approved