%I A143227
%S A143227 1,2,1,1,1,1,2,2,3,3,3,1,1,1,1,1,2,2,1,1,1,2,2,1,1,3,2,1,1,2,2,6,3,3,1,
%T A143227 1,1,2,1,1,1,1,6,3,8,3,2,3,2,3,1,1,4,3,10,2,1,1,2,3,1,3,4,2,2,9,7,2,2,
               4,
%U A143227 3,3,1,2,3,5,1,2,3,2,11,3,1,2,4,7,1,1,1,1,1,5,1,2,3,3,4,2,2,9,5,1,4,2,
               2
%N A143227 (Number of primes between n and 2n) - (number of primes between n^2 and 
               (n+1)^2), if > 0.
%C A143227 If the sequence is bounded (e.g., if it is finite), then Legendre's conjecture 
               is true: there is always a prime between n^2 and (n+1)^2, at least 
               for all sufficiently large n. This follows from the strong form of 
               Bertrand's postulate proved by Ramanujan (see A104272 Ramanujan primes).
%D A143227 M. Aigner and C. M. Ziegler, Proofs from The Book, Chapter 2, Springer, 
               NY, 2001.
%D A143227 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 
               5th ed., Oxford Univ. Press, 1989, p. 19.
%D A143227 S. Ramanujan, "A Proof of Bertrand's Postulate," J. Indian Math. Soc. 
               11 (1919) 181-182.
%D A143227 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.
%H A143227 T. D. Noe, <a href="b143227.txt">Table of n, a(n) for n=1..413</a>
%H A143227 T. Hashimoto, <a href="http://arxiv.org/abs/0807.3690"> On a certain 
               relation between Legendre's conjecture and Bertrand's postulate</
               a>
%H A143227 M. Hassani, <a href="http://arXiv.org/abs/math/0607096"> Counting primes 
               in the interval (n^2,(n+1)^2)</a>
%H A143227 T. D. Noe, <a href="http://www.sspectra.com/math/A143227.gif">Plot of 
               the points (A143226(n), A143227(n))</a>
%H A143227 J. Pintz, <a href="http://www.renyi.hu/~pintz/"> Landau's problems on 
               primes</a>
%H A143227 J. Sondow, <a href="http://mathworld.wolfram.com/RamanujanPrime.html"> 
               Ramanujan Prime in MathWorld</a>
%H A143227 J. Sondow and E. W. Weisstein, <a href="http://mathworld.wolfram.com/
               BertrandsPostulate.html"> Bertrand's Postulate in MathWorld</a>
%H A143227 E. W. Weisstein, <a href="http://mathworld.wolfram.com/LegendresConjecture.html"> 
               Legendre's Conjecture in MathWorld</a>
%F A143227 a(n) = |A143223(A143226(n))|
%e A143227 The 1st positive value of ((pi(2n) - pi(n)) - (pi((n+1)^2) - pi(n^2))) 
               is 1 (at n = 42), the 2nd is 2 (at n = 55) and the 3rd is 1 (at n 
               = 56), so a(1) = 1, a(2) = 2, a(3) = 1.
%t A143227 L={}; Do[ With[ {d=(PrimePi[2n]-PrimePi[n])-(PrimePi[(n+1)^2]-PrimePi[n^2])}, 
               If[d>0, L=Append[L,d]]], {n,0,1000}]; L
%Y A143227 Cf. A000720, A014085, A060715, A104272, A143223, A143224, A143225, A143226 
               = corresponding values of n.
%Y A143227 Sequence in context: A064823 A140225 A104758 this_sequence A026791 A080576 
               A083671
%Y A143227 Adjacent sequences: A143224 A143225 A143226 this_sequence A143228 A143229 
               A143230
%K A143227 nonn
%O A143227 1,2
%A A143227 Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 02 2008