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 A143224 Numbers n such that (number of primes between n^2 and (n+1)^2) = (number of primes between n and 2n). 9

%I

%S 0,9,36,37,46,49,85,102,107,118,122,127,129,140,157,184,194,216,228,

%T 360,365,377,378,406,416,487,511,571,609,614,672,733,767,806,813,863,

%U 869,916,923,950,978,988,1249,1279,1280,1385,1427,1437,1483,1539,1551,1690

%N Numbers n such that (number of primes between n^2 and (n+1)^2) = (number of primes between n and 2n).

%C The sequence gives the zeros in A143223. The number of primes in question is A143225(n).

%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.

%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.

%D S. Ramanujan, "A Proof of Bertrand's Postulate," J. Indian Math. Soc. 11 (1919) 181-182.

%D S. Ramanujan, Collected Papers of Srinivasa Ramanujan (G. H. Hardy, S. Aiyar, P. Venkatesvara and B. M. Wilson, eds.), Amer. Math. Soc., Providence, 2000, pp. 208-209. [From _Jonathan Sondow_, Aug 03 2008]

%H T. D. Noe, <a href="/A143224/b143224.txt">Table of n, a(n) for n=1..97</a> (no other n < 10^6)

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

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

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

%H J. Sondow, <a href="http://mathworld.wolfram.com/RamanujanPrime.html"> Ramanujan Prime in MathWorld</a> [From _Jonathan Sondow_, Aug 02 2008]

%H J. Sondow and E. W. Weisstein, <a href="http://mathworld.wolfram.com/BertrandsPostulate.html"> Bertrand's Postulate in MathWorld</a> [From _Jonathan Sondow_, Aug 02 2008]

%H E. W. Weisstein, <a href="http://mathworld.wolfram.com/LegendresConjecture.html"> Legendre's Conjecture in MathWorld</a> [From _Jonathan Sondow_, Aug 02 2008]

%H S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper24/page1.htm"> A Proof Of Bertrand's Postulate</a> [From _Jonathan Sondow_, Aug 03 2008]

%F A143223(n) = 0

%e There are the same number of primes (namely 3) between 9^2 and 10^2 as between 9 and 2*9, so 9 is a member.

%t L={}; Do[If[PrimePi[(n+1)^2]-PrimePi[n^2] == PrimePi[2n]-PrimePi[n], L=Append[L,n]], {n,0,2000}]; L

%Y See A000720, A014085, A060715, A143223, A143225, A143226.

%Y Cf. A104272, A143227. [From _Jonathan Sondow_, Aug 03 2008]

%K nonn

%O 1,2

%A _Jonathan Sondow_, Jul 31 2008

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