Search: id:A143225
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%I A143225
%S A143225 0,3,9,9,10,10,16,20,19,21,23,23,24,25,28,31,32,36,38,56,57,59,59,62,65,
%T A143225 71,75,84,88,88,96,102,107,115,116,119,120,126,125,129,132,132,163,168,
%U A143225 168,182,189,189,192,197,198,213,236
%N A143225 Number of primes between n^2 and (n+1)^2, if equal to the number of primes 
               between n and 2n.
%C A143225 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 A143225 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 A143225 M. Aigner and C. M. Ziegler, Proofs from The Book, Chapter 2, Springer, 
               NY, 2001.
%D A143225 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 
               5th ed., Oxford Univ. Press, 1989, p. 19.
%D A143225 S. Ramanujan, "A Proof of Bertrand's Postulate," J. Indian Math. Soc. 
               11 (1919) 181-182.
%H A143225 T. D. Noe, <a href="b143225.txt">Table of n, a(n) for n=1..97</a> (no 
               other n < 10^6)
%H A143225 T. Hashimoto, <a href="http://arxiv.org/abs/0807.3690"> On a certain 
               relation between Legendre's conjecture and Bertrand's postulate</
               a>
%H A143225 M. Hassani, <a href="http://arXiv.org/abs/math/0607096"> Counting primes 
               in the interval (n^2,(n+1)^2)</a>
%H A143225 J. Pintz, <a href="http://www.renyi.hu/~pintz/"> Landau's problems on 
               primes</a>
%H A143225 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 A143225 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 A143225 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 A143225 a(n) = A014085(A143224(n)) = A060715(A143224(n)) for n > 0
%e A143225 There are 3 primes between 9^2 and 10^2 and 3 primes between 9 and 2*9, 
               so 3 is a member.
%t A143225 L={}; Do[If[PrimePi[(n+1)^2]-PrimePi[n^2] == PrimePi[2n]-PrimePi[n], 
               L=Append[L,PrimePi[2n]-PrimePi[n]]], {n,0,2000}]; L
%Y A143225 See A000720, A014085, A060715, A143223, A143224, A143226.
%Y A143225 Cf. A104272, A143227. [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), 
               Aug 03 2008]
%Y A143225 Sequence in context: A004166 A110759 A063750 this_sequence A099720 A162349 
               A072404
%Y A143225 Adjacent sequences: A143222 A143223 A143224 this_sequence A143226 A143227 
               A143228
%K A143225 nonn
%O A143225 1,2
%A A143225 Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Jul 31 2008

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