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a(n) = A000040(A221056(n)). - Reinhard Zumkeller, Apr 15 2013

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

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

(Haskell)

a224363 = a000040 . a221056 -- Reinhard Zumkeller, Apr 15 2013

Cf. A061265, A014085.

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

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2, 5, 11, 17, 19, 29, 37, 41, 43, 53, 59, 67, 71, 73, 83, 89, 101, 103, 107, 109, 127, 131, 137, 149, 151, 157, 163, 173, 179, 181, 191, 197, 199, 211, 227, 229, 233, 239, 241, 257, 263, 269, 271, 277, 281, 293, 307, 311, 313, 331, 337, 347, 349, 353, 367, 373

Select[Prime@[Range@[60, ]], Floor@[Sqrt@[NextPrime@[# ]]] == Floor@[Sqrt@[# ]] &] (* Giovanni Resta, Apr 10 2013 *)

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Consecutive primes with Primes p such that there are no square squares between themp and the prime following p.

2, 5, 11, 17, 19, 29, 37, 41, 43, 53, 59, 67, 71, 73, 83, 89, 101, 103, 107, 109, 113, 127, 131, 137, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 227, 229, 233, 239, 241, 257, 263, 269, 271, 277, 281

On The Legendré 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.

Every square is between primes.

For n≤2, there is always a prime such that:

p< n^2 <p

and

p< (n+1)^2 <p

so

n^2<p<(n+1)^2<p

if

p<n^2=p<(n+1)^2 then p is between [n^2,(n+1)^2]

p<n^2≠p<(n+1)^2 then p is between [n^2, (n+1)^2]

if

p<n^2<(n+1)^2 and p>(n+1)^2>n^2

So we need that 'prime factors' (fp)

fp<n^2<(n+1)^2 and fp>(n+1)^2>n^2

then we have:

n+n^2+(n+1)=(n+1)^2

n+n^2<(n+1)^2

n+(n+1)<(n+1)^2

2n+1<(n+1)^2

then:

(fp)^2>(n+1)^2 (is not true)

so

knowing that for n≥3 there are no primes of the form n^2-1 and n^2 can not be a prime

The Legendré Conjecture is true for [(n^2+1),(n+1)^2-2]

5, is a term because there is are no square squares between the adjacent primes 5 and 7 or.

n^2< {5,7} < (n+1)^2 for n=2

Select[Prime@Range@60, Floor@Sqrt@NextPrime@# == Floor@Sqrt@# &] (* Giovanni Resta, Apr 10 2013 *)

Corrected and edited by Giovanni Resta, Apr 10 2013

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n^2< {5,7} < (n+1)^2 for n=2