A220274 a(n) is the smallest number such that for all N >= a(n) there are at least n primes between 9*N and 10*N.
2, 14, 23, 23, 34, 36, 57, 58, 60, 60, 77, 86, 100, 100, 102, 123, 149, 149, 149, 149, 187, 187, 200, 200, 200, 202, 209, 227, 234, 268, 269, 269, 270, 319, 319, 331, 332, 332, 333, 345, 347, 350, 350, 353, 359, 360, 377, 401, 421, 440, 449, 479, 479, 487, 491
Offset: 1
Keywords
Links
- Peter J. C. Moses, Table of n, a(n) for n = 1..3000
- N. Amersi, O. Beckwith, S. J. Miller, R. Ronan, and J. Sondow, Generalized Ramanujan primes, arXiv:1108.0475 [math.NT], 2011.
- N. Amersi, O. Beckwith, S. J. Miller, R. Ronan, and J. Sondow, Generalized Ramanujan primes, Combinatorial and Additive Number Theory, Springer Proc. in Math. & Stat., CANT 2011 and 2012, Vol. 101 (2014), 1-13
- Vladimir 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, and Peter J. C. Moses, On intervals (kn, (k+1)n) containing a prime for all n>1, arXiv:1212.2785 [math.NT], 2012.
Formula
a(n) <= ceiling(R_(10/9)(n)/10), where R_v(n) (v>1) are generalized Ramanujan numbers (see Shevelev's link). In particular, for n >= 1, {R_(10/9) (n)} = {127, 223, 227, 269, 349, 359, 569, 587, 593, 739, 809, 857, 991, 1009, ...}. Moreover, if R_(10/9)(n) == 1 (mod 10), then a(n) = ceiling(R_(10/9)(n)/10).