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

Author

T. D. Noe, May 11 2011

Keywords

• nonn
• easy

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

• Christian Axler, On the number of primes up to the n-th Ramanujan prime, arXiv:1711.04588 [math.NT], 2017.
• J. Sondow, Ramanujan primes and Bertrand's postulate, Amer. Math. Monthly 116 (2009) 630-635.
• 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.
• Shichun Yang and Alain Togbé, On the estimates of the upper and lower bounds of Ramanujan primes, Ramanujan J., online 14 August 2015, 1-11.

Crossrefs

Cf. A007395, A104272, A179196, A190414.

Formula

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