

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


LINKS



FORMULA



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]}];


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



