

A111252


Primes p such that the difference between the closest squares surrounding p is prime.


1



2, 3, 5, 7, 11, 13, 29, 31, 37, 41, 43, 47, 67, 71, 73, 79, 83, 89, 97, 127, 131, 137, 139, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 331, 337, 347, 349, 353, 359, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 541, 547, 557, 563
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OFFSET

1,1


COMMENTS

Conjecture: The number of terms in this sequence is infinite.
That there are infinitely many terms in this sequence would follow from the Legendre conjecture (one of the Landau problems  see the Weisstein link) that there is always a prime between n^2 and (n+1)^2. This is still an open problem.  Max Alekseyev, Apr 20 2006


LINKS

Table of n, a(n) for n=1..57.
Eric Weisstein's World of Mathematics, Landau's Problems


FORMULA

Let p be a prime number and r = floor(sqrt(p)). Then the closest surrounding squares of p are r^2 and (r+1)^2. So d = (r+1)^2  r^2 = 2r+1. If d is prime then list p.


EXAMPLE

29 is a prime number. 5^2 and 6^2 are the closest squares surrounding 29. Now the difference 3625 = 11 is prime so 29 is in the table.


MATHEMATICA

Clear[f, lst, p, n]; f[n_]:=IntegerPart[Sqrt[n]]; lst={}; Do[p=Prime[n]; If[PrimeQ[a=(f[p]+1)^2f[p]^2], AppendTo[lst, p]], {n, 6!}]; lst (* Vladimir Joseph Stephan Orlovsky, Aug 05 2009 *)
Select[Prime[Range@103], PrimeQ[2*Floor[Sqrt[#]]+1]&] (* Ivan N. Ianakiev, Jul 30 2015 *)


PROG

(PARI) surrsqpr(n) = { local(x, y, j, r, d); forprime(x=2, n, r=floor(sqrt(x)); d=r+r+1; if(isprime(d), print1(x, ", ") ) ) }


CROSSREFS

Sequence in context: A265885 A294994 A292205 * A181525 A082843 A162567
Adjacent sequences: A111249 A111250 A111251 * A111253 A111254 A111255


KEYWORD

easy,nonn


AUTHOR

Cino Hilliard, Nov 12 2005


STATUS

approved



