A290508 Radius of largest circle among 4 circles each of which kisses the other three externally, and for which the radii are mutually coprime integers.

12, 15, 24, 63, 69, 70, 72, 80, 105, 120, 120, 132, 153, 168, 198, 216, 231, 234, 238, 252, 264, 264, 270, 275, 280, 285, 290, 315, 336, 390, 392, 420, 429, 504, 504, 520, 539, 544, 567, 585, 595, 624, 627, 658, 658, 672, 672, 690, 693, 693, 696, 714, 715

Offset: 1

Author

Albert Lau, Aug 04 2017

Keywords

• nonn

Comments

Descartes's theorem: 4 kissing circles with radii a,b,c,d satisfy

(1/a + 1/b + 1/c + 1/d)^2 = 2 (1/a^2 + 1/b^2 + 1/c^2 + 1/d^2).

Examples

			The table gives the first 8 examples:
   a  b  c  d
  == == == ==
  12 12  3  1
  15 15 10  2
  24 24  2  1
  63 56 56  9
  69 46 23  6
  70 30 21  5
  72 45 45  8
  80 80 36  9
		

Links

• Ray Chandler, Table of n, a(n) for n = 1..153
• Graham, R. L.; Lagarias, J. C.; Mallows, C. L.; Wilks, A. R.; and Yan, C. H., Apollonian Circle Packings: Geometry and Group Theory I. The Apollonian Group., Discrete & Computational Geometry, 34 (2005), no. 4, 547-585.
• Graham, R. L.; Lagarias, J. C.; Mallows, C. L.; Wilks, A. R.; and Yan, C. H., Apollonian Circle Packings: Geometry and Group Theory II. Super-Apollonian Group and Integral Packings., Discrete & Computational Geometry, 35 (2006), no. 1, 1-36.
• Graham, R. L.; Lagarias, J. C.; Mallows, C. L.; Wilks, A. R.; and Yan, C. H., Apollonian Circle Packings: Geometry and Group Theory III. Higher Dimensions., Discrete & Computational Geometry, 35 (2006), no. 1, 37-72.
• Graham, R. L.; Lagarias, J. C.; Mallows, C. L.; Wilks, A. R.; and Yan, C. H., Apollonian Circle Packings: Number Theory, J. Number Theory, 100 (2003), 1-45.
• N. J. A. Sloane, Illustration for a(1)=12. Draw 4 mutually touching circles of radii 12, 12, 3, and 1. The curvatures (1/radius) are 1/12, 1/12, 1/3, and 1, or equivalently 1, 1, 4, and 12. Four such circles can be seen (in many ways) in the illustration, which is Fig. 1 of the Graham et al. (2006, Part II) article. Used with permission of the authors.
• Eric Weisstein's World of Mathematics, Soddy Circles.
• Eric Weisstein's World of Mathematics, Tangent Circles
• Wikipedia, Descartes' theorem

Crossrefs

Cf. A290931 (4 mutually tangent circles, 1 circle enclosing 3).

Programs

Mathematica

aMax = 150; (* WARNING: O(n^3) *)
Do[
     If[d // IntegerQ // Not, Continue[]];
     If[GCD[a, b, c, d] > 1, Continue[]];
     {a, b, c, d} // Sow;
     , {a, aMax}, {b, a}, {c, b},
       {d, {1/(1/a + 1/b + 1/c + 2 Sqrt[(a + b + c)/(a b c)])}}
] // Reap // Last // Last // TableForm

Extensions

Corrected (inserted 63, 72, 234, 275) and extended by Ray Chandler, Aug 06 2017

Edited by N. J. A. Sloane, Aug 16 2017