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

1,1

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

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, ...}.

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 *)

Crossrefs

Cf. A084140.

Sequence in context: A089728 A127987 A247543 * A049476 A216889 A334494

Adjacent sequences: A220265 A220266 A220267 * A220269 A220270 A220271

Keyword

nonn

Author

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

Status

approved