

A193761


0.25Ramanujan 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.


3



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
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OFFSET

1,1


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

Table of n, a(n) for n=1..56.
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), 113
V. Shevelev, Ramanujan and Labos primes, their generalizations and classifications of primes, arXiv:0909.0715 [math.NT], 20092011.
V. Shevelev, Ramanujan and Labos primes, their generalizations, and classifications of primes, J. Integer Seq. 15 (2012) Article 12.5.4


FORMULA

a(n) <= A104272(n).


CROSSREFS

Cf. A104272 (Ramanujan primes), A193880 (0.75Ramanujan primes), A164952, A290394 (first (1 + 1/n)Ramanujan prime).
Sequence in context: A262840 A215318 A186945 * A215355 A242752 A215813
Adjacent sequences: A193758 A193759 A193760 * A193762 A193763 A193764


KEYWORD

nonn


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


STATUS

approved



