

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.


17



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
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OFFSET

1,1


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


LINKS

Table of n, a(n) for n=1..54.
V. Shevelev, Ramanujan and Labos primes, their generalizations and classifications of primes, arXiv:0909.0715 [math.NT], 20092011.
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.


FORMULA

A080359(n) <= a(n) <= A104272(n) = R_n (Cf. A194184, A194186).
a(n)>p_(2*n1); a(n)~p_{2n} (Cf. properties of R_n in A104272 and the above comment).  Vladimir Shevelev, Aug 28 2011


EXAMPLE

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.


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]]&]] (* JeanFrançois Alcover, Nov 07 2018, after T. D. Noe in A104272 *)


CROSSREFS

Cf. A104272 (Ramanujan primes).
Sequence in context: A275030 A194598 A080359 * A103087 A302485 A135118
Adjacent sequences: A193504 A193505 A193506 * A193508 A193509 A193510


KEYWORD

nonn


AUTHOR

Vladimir Shevelev, Aug 18 2011


STATUS

approved



