%I A014085
%S A014085 0,2,2,2,3,2,4,3,4,3,5,4,5,5,4,6,7,5,6,6,7,7,7,6,9,8,7,8,9,8,8,10,9,10,
%T A014085 9,10,9,9,12,11,12,11,9,12,11,13,10,13,15,10,11,15,16,12,13,11,12,17,
%U A014085 13,16,16,13,17,15,14,16,15,15,17,13,21,15,15,17,17,18,22,14,18,23,13
%N A014085 Number of primes between n^2 and (n+1)^2.
%C A014085 Suggested by Legendre's conjecture (still open) that for n > 0 there 
               is always a prime between n^2 and (n+1)^2.
%C A014085 See the additional references and links mentioned in A143227. [From Jonathan 
               Sondow (jsondow(AT)alumni.princeton.edu), Aug 03 2008]
%C A014085 Legendre's conjecture may be written Pi((n+1)^2)-Pi(n^2) > 0 for all 
               positive n, where Pi(n) = A000720(n). - Jonathan Vos Post (jvospost3(AT)gmail.com), 
               Jul 30 2008 [Comment corrected by Jonathan Sondow (jsondow(AT)alumni.princeton.edu), 
               Aug 15 2008]
%C A014085 Legendre's conjecture can be generalized as follows: for all integers 
               n>0 and all real numbers k>K, there is a prime in the range n^k to 
               (n+1)^k. The constant K is conjectured to be log(127)/log(16). See 
               A143935. [From T. D. Noe (noe(AT)sspectra.com), Sep 05 2008]
%D A014085 J. R. Goldman, The Queen of Mathematics, 1998, p. 82.
%H A014085 T. D. Noe, <a href="b014085.txt">Table of n, a(n) for n = 0..10000</a>
%H A014085 Tsutomu Hashimoto, <a href="http://arxiv.org/abs/0807.3690">On a certain 
               relation between Legendre's conjecture and Bertrand's postulate</
               a>
%H A014085 M. Hassani, <a href="http://arXiv.org/abs/math.NT/0607096">Counting primes 
               in the interval (n^2, (n+1)^2)</a>
%H A014085 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               LegendresConjecture.html">Legendre's Conjecture</a>
%F A014085 a(n) is the number of occurrences of n in A000006 . - DELEHAM Philippe 
               (kolotoko(AT)wanadoo.fr), Dec 17 2003
%F A014085 Pi((n+1)^2)-Pi(n^2) = A000720((n+1)^2)-A000720(n^2). - Jonathan Vos Post 
               (jvospost3(AT)gmail.com), Jul 30 2008
%e A014085 a(17)=5 because between 17^2 and 18^2, i.e. 289 and 324 there are 5 primes 
               (which are 293, 307, 311, 313, 317).
%t A014085 Table[ct = PrimePi[(k + 1)^2] - PrimePi[k^2], {k, 0, 80}]. - Lei Zhou 
               (lzhou5(AT)emory.edu), Dec 01 2005
%Y A014085 Cf. A000006, A053000, A053001, A007491, A077766, A077767, A108954.
%Y A014085 Cf. A000720, A060715, A104272, A143223, A143224, A143225, A143226, A143227.
%Y A014085 Sequence in context: A126336 A134446 A125749 this_sequence A029210 A035433 
               A029199
%Y A014085 Adjacent sequences: A014082 A014083 A014084 this_sequence A014086 A014087 
               A014088
%K A014085 nonn,easy,nice
%O A014085 0,2
%A A014085 Jonathan Wild (jon(AT)sound.music.mcgill.ca)