A224363 - OEIS

%I #13 Apr 15 2013 10:07:35

%S 2,5,11,17,19,29,37,41,43,53,59,67,71,73,83,89,101,103,107,109,127,

%T 131,137,149,151,157,163,173,179,181,191,197,199,211,227,229,233,239,

%U 241,257,263,269,271,277,281,293,307,311,313,331,337,347,349,353,367,373

%N Primes p such that there are no squares between p and the prime following p.

%C Legendre's Conjecture states that there is a prime between n^2 and (n+1)^2 for every integer n > 0 and thus that between two adjacent primes there can be at most one square. As of April 2013, the conjecture is still unproved.

%C a(n) = A000040(A221056(n)). - _Reinhard Zumkeller_, Apr 15 2013

%H Reinhard Zumkeller, <a href="/A224363/b224363.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LegendresConjecture.html">Legendre's Conjecture</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Legendre%27s_conjecture">Legendre's conjecture</a>

%e 5 is a term because there are no squares between the adjacent primes 5 and 7.

%t Select[Prime[Range[60]], Floor[Sqrt[NextPrime[#]]] == Floor[Sqrt[#]] &] (* _Giovanni Resta_, Apr 10 2013 *)

%o (Haskell)

%o a224363 = a000040 . a221056  -- _Reinhard Zumkeller_, Apr 15 2013

%Y Cf. A061265, A014085.

%K nonn

%O 1,1

%A _César Aguilera_, Apr 04 2013

%E Corrected and edited by _Giovanni Resta_, Apr 10 2013