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

1,1

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

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).

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 *)

Crossrefs

Cf. A084140, A220268.

Sequence in context: A366915 A064105 A129516 * A143189 A056550 A074263

Adjacent sequences: A220266 A220267 A220268 * A220270 A220271 A220272

Keyword

nonn

Author

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

Status

approved