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.

11, 23, 47, 59, 83, 107, 131, 167, 227, 233, 239, 251, 263, 281, 347, 383, 401, 419, 431, 443, 479, 563, 587, 593, 641, 647, 653, 659, 719, 743, 809, 821, 839, 863, 941, 947, 971, 1019, 1049, 1061, 1091, 1151, 1187, 1217, 1223, 1259, 1283

Offset: 1

Author

Vladimir Shevelev and Peter J. C. Moses, Dec 18 2012

Keywords

• nonn

Comments

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

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

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.

Links

• Vladimir Shevelev, Charles R. Greathouse IV, Peter J. C. Moses, On intervals (kn, (k+1)n) containing a prime for all n>1, Journal of Integer Sequences, Vol. 16 (2013), Article 13.7.3. arXiv:1212.2785

Crossrefs

Cf. A104272, A185004, A185006, A185007.

Programs

Mathematica

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

Formula

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