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There appear to be at least n primes in the range (x-2*sqrt(x), x] for all x >= a(n).
4

%I #10 Mar 24 2017 00:47:53

%S 2,3,37,139,331,1409,1423,1427,2239,3163,3181,3511,6547,7433,7457,

%T 7487,10061,11777,11779,14401,18899,19081,19373,23537,24763,27617,

%U 27673,32027,32051,38113,43573,43579,47269,47279,50839,61463,88643,88651,88657,88729

%N There appear to be at least n primes in the range (x-2*sqrt(x), x] for all x >= a(n).

%C These terms exist only if a strong form of Legendre's conjecture that there is a prime between consecutive squares is true. Note that every term is prime. Sequence A189025 gives the number of primes in the range (x-2*sqrt(x), x]. The index of prime a(n), that is, primepi(a(n)), is approximately (5n)^2. These primes are generated in a manner similar to the Ramanujan primes (A104272).

%H T. D. Noe, <a href="/A189027/b189027.txt">Table of n, a(n) for n = 1..1000</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Legendre_conjecture">Legendre's conjecture</a>

%Y Cf. A189024, A189026.

%K nonn

%O 1,1

%A _T. D. Noe_, Apr 15 2011