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A274550 The curvature of smallest circle among 4 mutually tangent(externally) circles with integer curvature and primitive (share no common factor).

%I A274550 #22 Jul 10 2016 22:59:32
%S A274550 12,15,23,24,28,33,34,35,38,39,40,42,45,47,50,52,53,56,57,58,59,60,61,
%T A274550 62,63,63,64,66,69,71,72,72,73,76,77,77,79,80,81,82,82,83,84,84,85,86,
%U A274550 87,87,88,90,91,91,94,94,95,95,96,96,97,98,98,99,99
%N A274550 The curvature of smallest circle among 4 mutually tangent(externally) circles with integer curvature and primitive (share no common factor).
%C A274550 4 mutually tangent circles satisfy 2 (a^2 + b^2 + c^2 + d^2) = (a + b + c + d)^2 where a,b,c,d are the curvatures.
%H A274550 Wikipedia, <a href="https://en.wikipedia.org/wiki/Apollonian_gasket">Apollonian gasket</a>
%e A274550 a,  b,  c, d
%e A274550 12, 4,  1, 1
%e A274550 15, 3,  2, 2
%e A274550 23, 6,  3, 2
%e A274550 24, 12, 1, 1
%e A274550 28, 9,  4, 1
%t A274550 aMax = 100;
%t A274550 Do[
%t A274550     If[GCD[a, b, c] > 1, Continue[]];
%t A274550     d = a + b + c - 2 Sqrt[a b + a c + b c];
%t A274550     If[d // IntegerQ // Not, Continue[]];
%t A274550     (*{a,b,c,d}*)a // Sow;
%t A274550     , {a, aMax}
%t A274550     , {b, (2 a)/Sqrt[3] - a // Ceiling, (Sqrt[a] - 1)^2}
%t A274550     , {c,(a-b)^2/(4(a+b))//Ceiling,Min[b,(Sqrt[a]-Sqrt[b])^2-1//Ceiling]}
%t A274550 ] // Reap // Last // Last(*//TableForm*)
%t A274550 d =.;
%K A274550 nonn
%O A274550 1,1
%A A274550 _Albert Lau_, Jul 03 2016