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

12, 15, 23, 24, 28, 33, 34, 35, 38, 39, 40, 42, 45, 47, 50, 52, 53, 56, 57, 58, 59, 60, 61, 62, 63, 63, 64, 66, 69, 71, 72, 72, 73, 76, 77, 77, 79, 80, 81, 82, 82, 83, 84, 84, 85, 86, 87, 87, 88, 90, 91, 91, 94, 94, 95, 95, 96, 96, 97, 98, 98, 99, 99

Offset: 1

Author

Albert Lau, Jul 03 2016

Keywords

• nonn

Comments

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.

Examples

			a,  b,  c, d
12, 4,  1, 1
15, 3,  2, 2
23, 6,  3, 2
24, 12, 1, 1
28, 9,  4, 1
		

Links

• Wikipedia, Apollonian gasket

Programs

Mathematica

aMax = 100;
Do[
    If[GCD[a, b, c] > 1, Continue[]];
    d = a + b + c - 2 Sqrt[a b + a c + b c];
    If[d // IntegerQ // Not, Continue[]];
    (*{a,b,c,d}*)a // Sow;
    , {a, aMax}
    , {b, (2 a)/Sqrt[3] - a // Ceiling, (Sqrt[a] - 1)^2}
    , {c,(a-b)^2/(4(a+b))//Ceiling,Min[b,(Sqrt[a]-Sqrt[b])^2-1//Ceiling]}
] // Reap // Last // Last(*//TableForm*)
d =.;