A190414 primepi(R_m) <= i*primepi(R_j) for any factorization m=i*j if j >= a(i), where R_k is the k-th Ramanujan prime (A104272).

1, 2490, 567, 756, 425, 510, 70, 80, 90, 100, 110, 120, 130, 140, 150, 160, 170, 180, 190, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100

Offset

1,2

Comments

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

The conjecture has been proven for i > 38 and j > 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..50.

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

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*n.

Crossrefs

Cf. A104272, A179196, A190413.

Sequence in context: A255757 A254839 A231456 * A248548 A252315 A131523

Adjacent sequences: A190411 A190412 A190413 * A190415 A190416 A190417

Keyword

nonn

Author

John W. Nicholson, May 10 2011

Status

approved