%I #15 Sep 26 2021 11:09:37
%S 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,
%T 1,1,1,2,1,1,1,1,6,3,8,3,2,3,2,3,1,1,4,3,10,2,1,1,2,3,1,3,4,2,2,9,7,2,
%U 2,4,3,3,1,2,3,5,1,2,3,2,11,3,1,2,4,7,1,1,1,1,1,5,1,2,3,3,4,2,2,9,5,1,4,2,2
%N (Number of primes between n and 2n) - (number of primes between n^2 and (n+1)^2), if > 0.
%C 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 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, 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 T. D. Noe, <a href="/A143227/b143227.txt">Table of n, a(n) for n=1..413</a>
%H T. Hashimoto, <a href="http://arxiv.org/abs/0807.3690"> On a certain relation between Legendre's conjecture and Bertrand's postulate</a>, arXiv:0807.3690 [math.GM], 2008.
%H M. Hassani, <a href="http://arXiv.org/abs/math/0607096"> Counting primes in the interval (n^2,(n+1)^2)</a>, arXiv:math/0607096 [math.NT], 2006.
%H T. D. Noe, <a href="http://www.sspectra.com/math/A143227.gif">Plot of the points (A143226(n), A143227(n))</a>
%H J. Pintz, <a href="http://www.renyi.hu/~pintz/">Landau's problems on primes</a>
%H S. Ramanujan, <a href="http://ramanujan.sirinudi.org/Volumes/published/ram24.html">A proof of Bertrand's postulate</a>, J. Indian Math. Soc., 11 (1919), 181-182.
%H J. Sondow, <a href="http://mathworld.wolfram.com/RamanujanPrime.html">Ramanujan Prime in MathWorld</a>
%H J. Sondow and E. W. Weisstein, <a href="http://mathworld.wolfram.com/BertrandsPostulate.html">Bertrand's Postulate in MathWorld</a>
%H E. W. Weisstein, <a href="http://mathworld.wolfram.com/LegendresConjecture.html">Legendre's Conjecture in MathWorld</a>
%F a(n) = |A143223(A143226(n))|.
%e The first 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 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
%t Select[Table[(PrimePi[2n]-PrimePi[n])-(PrimePi[(n+1)^2]-PrimePi[n^2]),{n,1000}],#>0&] (* _Harvey P. Dale_, Jun 19 2019 *)
%Y Cf. A000720, A014085, A060715, A104272, A143223, A143224, A143225, A143226 = corresponding values of n.
%K nonn
%O 1,2
%A _Jonathan Sondow_, Aug 02 2008