Search: id:A143223
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%I A143223
%S A143223 0,2,1,1,1,1,2,1,2,0,1,1,1,2,1,2,2,1,2,2,3,2,1,1,3,2,1,1,2,2,1,3,2,3,1,
%T A143223 2,0,0,3,2,2,2,1,3,2,3,0,4,6,0,1,4,4,1,1,2,1,3,1,3,3,1,5,3,1,3,1,2,
%U A143223 4,1,6,1,1,4,4,4,7,1,3,8,2,5,3,5,1,0,5,5,1,2,3,2,1,5,3,3,2,3,4,1,2
%V A143223 0,2,1,1,1,1,2,1,2,0,1,1,1,2,1,2,2,1,2,2,3,2,1,1,3,2,1,1,2,2,1,3,2,3,1,
%W A143223 2,0,0,3,2,2,2,-1,3,2,3,0,4,6,0,1,4,4,1,1,-2,-1,3,-1,3,3,1,5,3,1,3,1,2,
%X A143223 4,-1,6,1,1,4,4,4,7,-1,3,8,-2,5,3,5,1,0,5,5,1,2,3,2,1,5,3,3,2,3,4,1,2
%N A143223 (Number of primes between n^2 and (n+1)^2) - (number of primes between 
               n and 2n).
%C A143223 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 A143223 Hashimoto's plot of (1 - a(n)) shows that |a(n)| is small compared to 
               n for n < 30,000.
%C A143223 Contribution from Jonathan Sondow (jsondow(AT)alumni.princeton.edu), 
               Aug 07 2008: (Start)
%C A143223 It appears that there are only a finite number of negative terms (see 
               A143226).
%C A143223 If the negative terms are bounded, then Legendre's conjecture is true, 
               at least for all sufficiently large n. This follows from the strong 
               form of Bertrand's postulate proved by Ramanujan (see A104272 Ramanujan 
               primes). (End)
%D A143223 M. Aigner and C. M. Ziegler, Proofs from The Book, Chapter 2, Springer, 
               NY, 2001.
%D A143223 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 
               5th ed., Oxford Univ. Press, 1989, p. 19.
%D A143223 S. Ramanujan, "A Proof of Bertrand's Postulate," J. Indian Math. Soc. 
               11 (1919) 181-182.
%D A143223 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 02 2008]
%H A143223 T. Hashimoto, <a href="http://arxiv.org/abs/0807.3690"> On a certain 
               relation between Legendre's conjecture and Bertrand's postulate</
               a>
%H A143223 M. Hassani, <a href="http://arXiv.org/abs/math/0607096"> Counting primes 
               in the interval (n^2,(n+1)^2)</a>
%H A143223 J. Pintz, <a href="http://www.renyi.hu/~pintz/"> Landau's problems on 
               primes</a>
%H A143223 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 A143223 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 A143223 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 A143223 T. D. Noe, <a href="http://www.sspectra.com/math/A143223.gif">Plot of 
               A143223(n) for n to 10^6</a> [From T. D. Noe (noe(AT)sspectra.com), 
               Aug 04 2008]
%H A143223 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 02 2008]
%F A143223 a(n) = A014085(n) - A060715(n) (for n > 0) = [pi((n+1)^2) - pi(n^2)] 
               - [pi(2n) - pi(n)] (for n > 1)
%e A143223 There are 4 primes between 6^2 and 7^2 and 2 primes between 6 and 2*6, 
               so a(6) = 4 - 2 = 2.
%e A143223 a(1) = 2 because there are two primes between 1^2 and 2^2 (namely, 2 
               and 3) and none between 1 and 2. [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), 
               Aug 07 2008]
%t A143223 L={0,2}; Do[L=Append[L,(PrimePi[(n+1)^2]-PrimePi[n^2]) - (PrimePi[2n]-PrimePi[n])], 
               {n,2,100}]; L
%Y A143223 See A000720, A014085, A060715, A143224, A143225, A143226.
%Y A143223 Negative terms are A143227. Cf. A104272 (Ramanujan primes). [From Jonathan 
               Sondow (jsondow(AT)alumni.princeton.edu), Aug 02 2008]
%Y A143223 Sequence in context: A083896 A084115 A080028 this_sequence A063993 A115722 
               A115721
%Y A143223 Adjacent sequences: A143220 A143221 A143222 this_sequence A143224 A143225 
               A143226
%K A143223 sign
%O A143223 0,2
%A A143223 Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Jul 31 2008
%E A143223 Corrected by Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 07 
               2008, Aug 09 2008

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