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%I #34 Nov 30 2019 01:28:32
%S 2,7,17,24,25,38,41,58,59,64,65,73,95,97,103,106,107,108,138,143,143,
%T 157,169,169,174,179,182,214,227,238,239,242,248,267,267,268,269,269,
%U 329,330,333,336,343,348,353,368,379,379,383,389,392,432,437,437,444
%N a(n) is the smallest number, such that for all N >= a(n) there are at least n primes between 5*N and 6*N.
%H Peter J. C. Moses, <a href="/A220273/b220273.txt">Table of n, a(n) for n = 1..3000</a>
%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 2011.
%H N. Amersi, O. Beckwith, S. J. Miller, R. Ronan, J. Sondow, <a href="http://link.springer.com/chapter/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://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
%H Vladimir Shevelev, Charles R. Greathouse IV, Peter J. C. Moses, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Moses/moses1.html">On intervals (kn, (k+1)n) containing a prime for all n>1</a>, Journal of Integer Sequences, Vol. 16 (2013), Article 13.7.3. <a href="http://arxiv.org/abs/1212.2785">arXiv:1212.2785</a>
%F a(n) <= ceiling(R_(6/5)(n)/6), where R_v(n) (v>1) are generalized Ramanujan numbers (see Shevelev's link). In particular, for n >= 1, {R_(6/5)(n)}={29, 59, 137, 139, 149, 223, 241, 347, 353, 383, 389, 563, 569, 593, ...}. Moreover, if R_(6/5)(n) == 1 (mod 6), then a(n) = ceiling(R_(6/5)(n)/6).
%Y Cf. A084140, A220268, A220269.
%K nonn
%O 1,1
%A _Vladimir Shevelev_, _Charles R Greathouse IV_ and _Peter J. C. Moses_, Dec 09 2012
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