A220269 a(n) is the smallest number, such that for all N >= a(n) there are at least n primes between 3*N and 4*N.

2, 8, 11, 17, 26, 38, 40, 41, 48, 57, 68, 68, 70, 87, 96, 100, 108, 109, 110, 115, 136, 149, 151, 161, 161, 169, 176, 178, 184, 206, 208, 227, 235, 236, 242, 255, 259, 260, 263, 272, 297, 299, 305, 320, 356, 358, 359, 371, 371, 372, 378, 386, 389, 392, 400

Offset: 1

Author

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

Keywords

• nonn

Links

• Peter J. C. Moses, Table of n, a(n) for n = 1..3000
• 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, A220268.

Programs

Mathematica

nn = 60; t = Table[PrimePi[4 n] - PrimePi[3 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}]] (* _Michael B. Porter, after A220268 program by T. D. Noe, Feb 09 2013 *)

Formula

a(n) <= ceiling(R_(4/3)(n)/4), where R_v(n) (v>1) are generalized Ramanujan numbers (see Shevelev's link). In particular, for n >= 1, {R_(4/3)(n)}={11, 29, 59, 67, 101, 149, 157, 163, 191, 227, 269, 271, 307, 379, ...}. Moreover, if R_(4/3)(n) == 1 (mod 4), then a(n) = ceiling(R_(4/3)(n)/4).