A220268 a(n) is the smallest number, such that for N >= a(n) there are at least n primes between 2*N and 3*N.

2, 5, 13, 14, 23, 25, 33, 43, 46, 58, 60, 61, 71, 77, 80, 88, 103, 104, 116, 123, 127, 144, 145, 148, 150, 160, 163, 181, 188, 196, 200, 203, 206, 214, 218, 237, 247, 253, 263, 266, 270, 275, 276, 287, 313, 323, 333, 340, 344, 347, 350, 354, 363, 365, 388

Offset: 1

Author

Vladimir Shevelev, Charles R Greathouse IV and Peter J. C. Moses, Dec 09 2012

Keywords

• nonn

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• N. Amersi, O. Beckwith, S. J. Miller, R. Ronan, J. Sondow, Generalized Ramanujan primes, arXiv 2011.
• N. Amersi, O. Beckwith, S. J. Miller, R. Ronan, J. Sondow, Generalized Ramanujan primes, Combinatorial and Additive Number Theory, Springer Proc. in Math. & Stat., CANT 2011 and 2012, Vol. 101 (2014), 1-13
• V. Shevelev, Ramanujan and Labos primes, their generalizations, and classifications of primes, J. Integer Seq. 15 (2012) Article 12.5.4
• 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. A084140.

Programs

Mathematica

nn = 60; t = Table[PrimePi[3 n] - PrimePi[2 n], {n, 10*nn}]; Join[{2}, Table[s = Flatten[Position[t, ?(# > n - 1 &)]]; i = Length[s]; While[i > 1 && s[[i]] - s[[i - 1]] == 1, i--]; s[[i]], {n, 2, nn}]] (* _T. D. Noe, Dec 12 2012 *)

Formula

a(1) = 2; for n >= 2, a(n) = ceiling(R_(3/2)(n)/3), where R_v(n) (v>1) are generalized Ramanujan numbers (see Shevelev's link). In particular, for n >= 1, {R_(3/2)(n)} = {2, 13, 37, 41, 67, 73, 97, 127, 137, 173, 179, 181, 211, 229, 239, ...}.