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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

%I #38 Feb 11 2021 01:25:36

%S 2,11,24,37,38,39,50,96,96,96,96,97,97,125,125,132,178,178,178,179,

%T 179,180,213,221,222,222,224,235,235,242,282,283,307,309,310,360,360,

%U 361,362,366,367,367,377,377,377,421,422,458,458,502,503,504

%N 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.

%H Peter J. C. Moses, <a href="/A220281/b220281.txt">Table of n, a(n) for n = 1..3000</a>

%H N. Amersi, O. Beckwith, S. J. Miller, R. Ronan, and 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, and 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 Vladimir 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, Сharles R. Greathouse IV, and Peter J. C. Moses, <a href="http://arxiv.org/abs/1212.2785">On intervals (kn, (k+1)n) containing a prime for all n>1</a>, arXiv:1212.2785 [math.NT], 2012.

%F 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).

%Y Cf. A084140, A220268, A220269, A220273, A220274.

%K nonn

%O 1,1

%A _Vladimir Shevelev_, _Charles R Greathouse IV_ and _Peter J. C. Moses_, Dec 09 2012

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