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0.75-Ramanujan primes R_{0.75,n}: a(n) is the smallest number such that for all x >= a(n), we have pi(x) - pi(0.75x) >= n, where pi(x) is the number of primes <= x.
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%I #32 Aug 02 2017 05:40:31

%S 11,29,59,67,101,149,157,163,191,227,269,271,307,379,383,419,431,433,

%T 443,457,563,593,601,641,643,673,701,709,733,827,829,907,937,947,971,

%U 1019,1033,1039,1051,1087,1187,1193,1217,1277,1427,1429,1433,1481,1483,1487

%N 0.75-Ramanujan primes R_{0.75,n}: a(n) is the smallest number such that for all x >= a(n), we have pi(x) - pi(0.75x) >= n, where pi(x) is the number of primes <= x.

%C See comment to A193761. - Vladimir Shevelev, Aug 18 2011

%C See additional comments and links in A290394. - _Jonathan Sondow_, Aug 01 2017

%H N. Amersi, O. Beckwith, S. J. Miller, R. Ronan, J. Sondow, <a href="http://arxiv.org/abs/1108.0475">Generalized Ramanujan primes</a>, arXiv:1108.0475 [math.NT], 2011.

%H N. Amersi, O. Beckwith, S. J. Miller, R. Ronan, J. Sondow, <a href="https://doi.org/10.1007/978-1-4939-1601-6_1">Generalized Ramanujan primes</a>, Combinatorial and Additive Number Theory, Springer Proc. in Math. & Stat., CANT 2011 and 2012, Vol. 101 (2014), 1-13

%H V. Shevelev, <a href="http://arxiv.org/abs/0909.0715">Ramanujan and Labos primes, their generalizations and classifications of primes</a>, arXiv:0909.0715 [math.NT], 2009-2011.

%H V. Shevelev, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL15/Shevelev/shevelev19.html">Ramanujan and Labos primes, their generalizations, and classifications of primes</a>, J. Integer Seq. 15 (2012) Article 12.5.4

%F a(n) >= A104272(n).

%e a(1) = A290394(3) = 11. - _Jonathan Sondow_, Aug 01 2017

%Y Cf. A104272 (Ramanujan primes), A193761 (0.25-Ramanujan primes), A164952, A290394 (first (1 + 1/n)-Ramanujan prime).

%K nonn

%O 1,1

%A _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 07 2011