%I A143226
%S A143226 42,55,56,58,69,77,80,119,136,137,143,145,149,156,174,177,178,188,219,
%T A143226 225,232,247,253,254,257,261,263,297,306,310,325,327,331,335,339,341,
%U A143226 344,356,379,395,402,410,418,421,425,433,451,485,500
%N A143226 Numbers n such that there are more primes between n and 2n than between 
               n^2 and (n+1)^2.
%C A143226 Legendre's conjecture (still open) says there is always a prime between 
               n^2 and (n+1)^2. Bertrand's postulate (actually a theorem due to 
               Chebychev) says there is always a prime between n and 2n.
%C A143226 It appears that this sequence is finite; searching up to 10^5, the last 
               n appears to be 48717. [From T. D. Noe (noe(AT)sspectra.com), Aug 
               01 2008]
%C A143226 If the sequence is finite, then, by Bertrand's postulate, Legendre's 
               conjecture is true, at least for all sufficiently large n. [From 
               Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 02 2008]
%C A143226 No other n <= 10^6. The plot of A143223 shows that it is quite likely 
               that there are no additional terms. [From T. D. Noe (noe(AT)sspectra.com), 
               Aug 04 2008]
%C A143226 See the additional reference and link to Ramanujan's work mentioned in 
               A143223. [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), 
               Aug 03 2008]
%D A143226 M. Aigner and C. M. Ziegler, Proofs from The Book, Chapter 2, Springer, 
               NY, 2001.
%D A143226 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 
               5th ed., Oxford Univ. Press, 1989, p. 19.
%D A143226 S. Ramanujan, "A Proof of Bertrand's Postulate," J. Indian Math. Soc. 
               11 (1919) 181-182.
%H A143226 T. D. Noe, <a href="b143226.txt">Table of n, a(n) for n=1..413</a>
%H A143226 T. Hashimoto, <a href="http://arxiv.org/abs/0807.3690"> On a certain 
               relation between Legendre's conjecture and Bertrand's postulate</
               a>
%H A143226 M. Hassani, <a href="http://arXiv.org/abs/math/0607096"> Counting primes 
               in the interval (n^2,(n+1)^2)</a>
%H A143226 J. Pintz, <a href="http://www.renyi.hu/~pintz/"> Landau's problems on 
               primes</a>
%H A143226 J. Sondow, <a href="http://mathworld.wolfram.com/RamanujanPrime.html"> 
               Ramanujan Prime in MathWorld</a> [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), 
               Aug 02 2008]
%H A143226 J. Sondow and E. W. Weisstein, <a href="http://mathworld.wolfram.com/
               BertrandsPostulate.html"> Bertrand's Postulate in MathWorld</a> [From 
               Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 02 2008]
%H A143226 E. W. Weisstein, <a href="http://mathworld.wolfram.com/LegendresConjecture.html"> 
               Legendre's Conjecture in MathWorld</a> [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), 
               Aug 02 2008]
%F A143226 A143223(n) < 0
%e A143226 There are 10 primes between 42 and 2*42, but only 9 primes between 42^2 
               and 43^2, so 42 is a member.
%t A143226 L={}; Do[If[PrimePi[(n+1)^2]-PrimePi[n^2] < PrimePi[2n]-PrimePi[n], L=Append[L,
               n]], {n,0,500}]; L
%Y A143226 See A000720, A014085, A060715, A143223, A143224, A143225.
%Y A143226 Cf. A104272, A143227. [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), 
               Aug 03 2008]
%Y A143226 Sequence in context: A125009 A008886 A029695 this_sequence A043136 A039313 
               A043916
%Y A143226 Adjacent sequences: A143223 A143224 A143225 this_sequence A143227 A143228 
               A143229
%K A143226 nonn
%O A143226 1,1
%A A143226 Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Jul 31 2008