%I A111205
%S A111205 14,18,21,24,33,39,50,51,54,55,57,60,63,68,95,102,105,108,111,112,114,
%T A111205 117,119,120,138,145,150,155,160,165,171,174,177,180,183,186,189,192,
%U A111205 195,203,248,258,261,264,267,270,273,275,276,279,282,285,286,288,290
%N A111205 Square-free numbers n such that the difference between the closest squares 
               surrounding n and n have a common divisor greater than 1.
%C A111205 Conjecture: The number of terms in this sequence is infinite.
%F A111205 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 A111205 14 is a squarefree composite number. 3^2 and 4^2 are the closest squares
%e A111205 surrounding 14. So the difference, 16-9 = 7 and 14 have a common divisor
%e A111205 greater than 1 namely 7, so 14 is the first entry in the table.
%o A111205 (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",") ) )) }
%Y A111205 Sequence in context: A060504 A052026 A118499 this_sequence A097324 A051419 
               A000053
%Y A111205 Adjacent sequences: A111202 A111203 A111204 this_sequence A111206 A111207 
               A111208
%K A111205 easy,nonn
%O A111205 6,1
%A A111205 Cino Hilliard (hillcino368(AT)gmail.com), Nov 12 2005