A193761 0.25-Ramanujan primes R_{0.25,n}: a(n) is the smallest number such that for any x >= a(n), we have pi(x) - pi(0.25x) >= n, where pi(x) is the number of primes <= x.

2, 3, 5, 13, 17, 29, 31, 37, 41, 53, 59, 61, 71, 79, 83, 97, 101, 103, 107, 127, 131, 137, 149, 151, 157, 173, 179, 191, 193, 197, 199, 223, 227, 229, 239, 251, 257, 269, 271, 277, 293, 307, 311, 317, 337, 347, 349, 359, 367, 373, 379, 389, 397, 419, 431, 439

Offset: 1

Author

Nadine Amersi, Olivia Beckwith (obeckwith(AT)gmail.com), Steven J. Miller (Steven.J.Miller(AT)williams.edu), Ryan Ronan (ronan2(AT)cooper.edu), Jonathan Sondow Aug 04 2011

Keywords

• nonn

Comments

Generalized Ramanujan primes with the parameter k have been introduced for the first time by Vladimir Shevelev in comment to A164952 from Sep 01 2009 (see also his comment to A104272 from the same date). Amersi et al. give the same definition with the parameter c=1/k in their cited paper. - Vladimir Shevelev, Aug 18 2011

See additional comments and links in A290394. - Jonathan Sondow, Aug 01 2017

Links

• N. Amersi, O. Beckwith, S. J. Miller, R. Ronan, J. Sondow, Generalized Ramanujan primes, arXiv:1108.0475 [math.NT], 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, arXiv:0909.0715 [math.NT], 2009-2011.
• V. Shevelev, Ramanujan and Labos primes, their generalizations, and classifications of primes, J. Integer Seq. 15 (2012) Article 12.5.4

Crossrefs

Cf. A104272 (Ramanujan primes), A193880 (0.75-Ramanujan primes), A164952, A290394 (first (1 + 1/n)-Ramanujan prime).

Formula

a(n) <= A104272(n).