Search: id:A143224
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%I A143224
%S A143224 0,9,36,37,46,49,85,102,107,118,122,127,129,140,157,184,194,216,228,360,
%T A143224 365,377,378,406,416,487,511,571,609,614,672,733,767,806,813,863,869,
%U A143224 916,923,950,978,988,1249,1279,1280,1385,1427,1437,1483,1539,1551,1690
%N A143224 Numbers n such that (number of primes between n^2 and (n+1)^2) = (number 
               of primes between n and 2n).
%C A143224 The sequence gives the zeros in A143223. The number of primes in question 
               is A143225(n).
%C A143224 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.
%D A143224 M. Aigner and C. M. Ziegler, Proofs from The Book, Chapter 2, Springer, 
               NY, 2001.
%D A143224 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 
               5th ed., Oxford Univ. Press, 1989, p. 19.
%D A143224 S. Ramanujan, "A Proof of Bertrand's Postulate," J. Indian Math. Soc. 
               11 (1919) 181-182.
%D A143224 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 (jsondow(AT)alumni.princeton.edu), 
               Aug 03 2008]
%H A143224 T. D. Noe, <a href="b143224.txt">Table of n, a(n) for n=1..97</a> (no 
               other n < 10^6)
%H A143224 T. Hashimoto, <a href="http://arxiv.org/abs/0807.3690"> On a certain 
               relation between Legendre's conjecture and Bertrand's postulate</
               a>
%H A143224 M. Hassani, <a href="http://arXiv.org/abs/math/0607096"> Counting primes 
               in the interval (n^2,(n+1)^2)</a>
%H A143224 J. Pintz, <a href="http://www.renyi.hu/~pintz/"> Landau's problems on 
               primes</a>
%H A143224 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 A143224 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 A143224 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]
%H A143224 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 (jsondow(AT)alumni.princeton.edu), Aug 03 2008]
%F A143224 A143223(n) = 0
%e A143224 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 A143224 L={}; Do[If[PrimePi[(n+1)^2]-PrimePi[n^2] == PrimePi[2n]-PrimePi[n], 
               L=Append[L,n]], {n,0,2000}]; L
%Y A143224 See A000720, A014085, A060715, A143223, A143225, A143226.
%Y A143224 Cf. A104272, A143227. [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), 
               Aug 03 2008]
%Y A143224 Sequence in context: A118414 A137628 A020297 this_sequence A068810 A077115 
               A073946
%Y A143224 Adjacent sequences: A143221 A143222 A143223 this_sequence A143225 A143226 
               A143227
%K A143224 nonn
%O A143224 1,2
%A A143224 Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Jul 31 2008

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