|
%I #15 Sep 26 2021 11:09:31
%S 0,3,9,9,10,10,16,20,19,21,23,23,24,25,28,31,32,36,38,56,57,59,59,62,
%T 65,71,75,84,88,88,96,102,107,115,116,119,120,126,125,129,132,132,163,
%U 168,168,182,189,189,192,197,198,213,236
%N Number of primes between n^2 and (n+1)^2, if equal to the number of primes between n and 2n.
%C 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 Chebyshev) says there is always a prime between n and 2n.
%C See the additional reference and link to Ramanujan's work mentioned in A143223. [_Jonathan Sondow_, Aug 03 2008]
%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.
%H T. D. Noe, <a href="/A143225/b143225.txt">Table of n, a(n) for n=1..97</a> (no other n < 10^6)
%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 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) = A014085(A143224(n)) = A060715(A143224(n)) for n > 0.
%e There are 3 primes between 9^2 and 10^2 and 3 primes between 9 and 2*9, so 3 is a member.
%t 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 See A000720, A014085, A060715, A143223, A143224, A143226.
%Y Cf. A104272, A143227. [_Jonathan Sondow_, Aug 03 2008]
%K nonn
%O 1,2
%A _Jonathan Sondow_, Jul 31 2008
|
