%I #9 Oct 01 2013 17:58:20
%S 14,18,21,24,33,39,50,51,54,55,57,60,63,68,95,102,105,108,111,112,114,
%T 117,119,120,138,145,150,155,160,165,171,174,177,180,183,186,189,192,
%U 195,203,248,258,261,264,267,270,273,275,276,279,282,285,286,288,290
%N Squarefree numbers n such that the difference between the closest squares surrounding n and n have a common divisor greater than 1.
%C Conjecture: The number of terms in this sequence is infinite.
%F Let n be a squarefree composite number and r = floor(sqrt(n)). Then the closest surrounding squares of n are r^2 and (r+1)^2. So d = (r+1)^2 - r^2 = 2r+1. If gcd(n, d) > 1 then list n.
%e 14 is a squarefree composite number. 3^2 and 4^2 are the closest squares surrounding 14. So the difference, 16-9 = 7 and 14 have a common divisor greater than 1 namely 7, so 14 is the first entry in the table.
%o (PARI) surrsqgcd(n) = { local(x,y,j,r,d); for(x=1,n, if(!issquare(x)&!isprime(x), r=floor(sqrt(x)); d=r+r+1; if(gcd(x,d) > 1, print1(x",") ) )) }
%K easy,nonn
%O 6,1
%A _Cino Hilliard_, Nov 12 2005