A193507 Ramanujan primes of the second kind: a(n) is the smallest prime such that if prime x >= a(n), then pi(x) - pi(x/2) >= n, where pi(x) is the number of primes <= x.

2, 3, 13, 19, 31, 43, 53, 61, 71, 73, 101, 103, 109, 131, 151, 157, 173, 181, 191, 229, 233, 239, 241, 251, 269, 271, 283, 311, 313, 349, 353, 373, 379, 409, 419, 421, 433, 439, 443, 463, 491, 499, 509, 571, 577, 593, 599, 601, 607, 613, 643, 647, 653, 659

Offset: 1

Author

Vladimir Shevelev, Aug 18 2011

Keywords

• nonn

Comments

Apparently A168425 and the 2. - R. J. Mathar, Aug 25 2011

An odd prime p is in the sequence iff the previous prime is Ramanujan. The Ramanujan primes and the Ramanujan primes of the second kind are the mutually wrapping up sequences: a(1)<=R_1<=a(2)<=R_2<=a(3)<=R_3<=.... . - Vladimir Shevelev, Aug 29 2011

All terms of the sequence are in A194598. - Vladimir Shevelev, Aug 30 2011

Examples

			Since R_2=11 (see A104272), then for x >= 11, we have pi(x) - pi(x/2) >= 2. However, if to consider only prime x, then we see that, for x=7,5,3, pi(x) - pi(x/2)= 2, but pi(2) - pi(1)= 1. Therefore, already for prime x>=3, we have pi(x) - pi(x/2) >= 2. Thus a(2)=3.
		

Links

• V. Shevelev, Ramanujan and Labos primes, their generalizations and classifications of primes, arXiv:0909.0715 [math.NT], 2009-2011.
• J. Sondow, Ramanujan Prime in MathWorld
• J. Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, arXiv:1105.2249 [math.NT], 2011; J. Integer Seq. 14 (2011) Article 11.6.2.

Crossrefs

Cf. A104272 (Ramanujan primes).

Programs

Mathematica

nn = 120; (* nn=120 returns 54 terms *)
R = Table[0, {nn}]; s = 0;
Do[If[PrimeQ[k], s++]; If[PrimeQ[k/2], s--]; If[s < nn, R[[s + 1]] = k], {k, Prime[3 nn]}];
A104272 = R + 1;
Join[{2}, Select[Prime[Range[nn]], MemberQ[A104272, NextPrime[#, -1]]&]] (* Jean-François Alcover, Nov 07 2018, after T. D. Noe in A104272 *)

Formula

A080359(n) <= a(n) <= A104272(n) = R_n (Cf. A194184, A194186).

a(n)>p_(2*n-1); a(n)~p_{2n} (Cf. properties of R_n in A104272 and the above comment). - Vladimir Shevelev, Aug 28 2011