A220281 a(n) is the smallest number, such that for all N >= a(n) there are at least n primes between 14*N and 15*N.
2, 11, 24, 37, 38, 39, 50, 96, 96, 96, 96, 97, 97, 125, 125, 132, 178, 178, 178, 179, 179, 180, 213, 221, 222, 222, 224, 235, 235, 242, 282, 283, 307, 309, 310, 360, 360, 361, 362, 366, 367, 367, 377, 377, 377, 421, 422, 458, 458, 502, 503, 504
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_(15/14)(n)/15), where R_v(n) (v>1) are generalized Ramanujan numbers (see Shevelev's link). In particular, for n >= 1, {R_(15/14)(n)}={127, 307, 347, 563, 569, 733, 1423, 1427, 1429, 1433, 1439, 1447, ...}. Moreover, if R_(15/14)(n) == 1 or 2 (mod 10), then a(n) = ceiling(R_(15/14)(n)/15).