A185005 - OEIS

A185005

Ramanujan primes R_(3,2)(n): a(n) is the smallest number such that if x >= a(n), then pi_(3,2)(x) - pi_(3,2)(x/2) >= n, where pi_(3,2)(x) is the number of primes==2 (mod 3) <= x.

%I #41 Aug 01 2013 09:39:32

%S 11,23,47,59,83,107,131,167,227,233,239,251,263,281,347,383,401,419,

%T 431,443,479,563,587,593,641,647,653,659,719,743,809,821,839,863,941,

%U 947,971,1019,1049,1061,1091,1151,1187,1217,1223,1259,1283

%N Ramanujan primes R_(3,2)(n): a(n) is the smallest number such that if x >= a(n), then pi_(3,2)(x) - pi_(3,2)(x/2) >= n, where pi_(3,2)(x) is the number of primes==2 (mod 3) <= x.

%C All terms are primes==2 (mod 3).

%C For the definition of generalized Ramanujan numbers, see Section 6 of the Shevelev, Greathouse, & Moses link.

%C We conjecture that for all n >= 1, a(n) <= A104272(3*n). This conjecture is based on observation that, if interval (x/2, x] contains >= 3*n primes, then at least n of them are of the form 3*k+2.

%H Vladimir Shevelev, Charles R. Greathouse IV, Peter J. C. Moses, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Moses/moses1.html">On intervals (kn, (k+1)n) containing a prime for all n>1</a>, Journal of Integer Sequences, Vol. 16 (2013), Article 13.7.3. <a href="http://arxiv.org/abs/1212.2785">arXiv:1212.2785</a>

%F lim(a(n)/prime(4*n)) = 1 as n tends to infinity.

%t Table[1 + NestWhile[#1 - 1 &, A104272[[3 k]], Count[Mod[Select[Range@@{Floor[#1/2 + 1], #1}, PrimeQ], 3], 2] >= k &], {k, 1, 10}]

%Y Cf. A104272, A185004, A185006, A185007.

%K nonn

%O 1,1

%A _Vladimir Shevelev_ and _Peter J. C. Moses_, Dec 18 2012