%I #18 Sep 21 2021 19:51:52

%S 42,55,56,58,69,77,80,119,136,137,143,145,149,156,174,177,178,188,219,

%T 225,232,247,253,254,257,261,263,297,306,310,325,327,331,335,339,341,

%U 344,356,379,395,402,410,418,421,425,433,451,485,500

%N Numbers n such that there are more primes between n and 2n than between n^2 and (n+1)^2.

%C 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 Chebyshev) says there is always a prime between n and 2n.

%C It appears that this sequence is finite; searching up to 10^5, the last n appears to be 48717. [_T. D. Noe_, Aug 01 2008]

%C If the sequence is finite, then, by Bertrand's postulate, Legendre's conjecture is true for sufficiently large n. - _Jonathan Sondow_, Aug 02 2008

%C No other n <= 10^6. The plot of A143223 shows that it is quite likely that there are no additional terms. - _T. D. Noe_, Aug 04 2008

%C See the additional reference and link to Ramanujan's work mentioned in A143223. - _Jonathan Sondow_, Aug 03 2008

%D M. Aigner and C. M. Ziegler, Proofs from The Book, Chapter 2, Springer, NY, 2001.

%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 5th ed., Oxford Univ. Press, 1989, p. 19.

%H T. D. Noe, <a href="/A143226/b143226.txt">Table of n, a(n) for n = 1..413</a>

%H T. Hashimoto, <a href="http://arxiv.org/abs/0807.3690">On a certain relation between Legendre's conjecture and Bertrand's postulate</a>, arXiv:0807.3690 [math.GM], 2008.

%H M. Hassani, <a href="http://arXiv.org/abs/math/0607096">Counting primes in the interval (n^2,(n+1)^2)</a>, arXiv:math/0607096 [math.NT], 2006.

%H J. Pintz, <a href="http://www.renyi.hu/~pintz/">Landau's problems on primes</a>

%H S. Ramanujan, <a href="http://ramanujan.sirinudi.org/Volumes/published/ram24.html">A proof of Bertrand's postulate</a>, J. Indian Math. Soc., 11 (1919), 181-182.

%H J. Sondow, <a href="http://mathworld.wolfram.com/RamanujanPrime.html">Ramanujan Prime in MathWorld</a>

%H J. Sondow and E. W. Weisstein, <a href="http://mathworld.wolfram.com/BertrandsPostulate.html">Bertrand's Postulate in MathWorld</a>

%H E. W. Weisstein, <a href="http://mathworld.wolfram.com/LegendresConjecture.html">Legendre's Conjecture in MathWorld</a>

%F A143223(n) < 0.

%e 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 L={}; Do[If[PrimePi[(n+1)^2]-PrimePi[n^2] < PrimePi[2n]-PrimePi[n], L=Append[L,n]], {n,0,500}]; L

%Y See A000720, A014085, A060715, A143223, A143224, A143225, A104272, A143227.

%K nonn

%O 1,1

%A _Jonathan Sondow_, Jul 31 2008