cp's OEIS Frontend

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.

%I A290508 #60 Feb 16 2025 08:33:49
%S A290508 12,15,24,63,69,70,72,80,105,120,120,132,153,168,198,216,231,234,238,
%T A290508 252,264,264,270,275,280,285,290,315,336,390,392,420,429,504,504,520,
%U A290508 539,544,567,585,595,624,627,658,658,672,672,690,693,693,696,714,715
%N 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.
%C A290508 Descartes's theorem: 4 kissing circles with radii a,b,c,d satisfy
%C A290508 (1/a + 1/b + 1/c + 1/d)^2 = 2 (1/a^2 + 1/b^2 + 1/c^2 + 1/d^2).
%H A290508 Ray Chandler, <a href="/A290508/b290508.txt">Table of n, a(n) for n = 1..153</a>
%H A290508 Graham, R. L.; Lagarias, J. C.; Mallows, C. L.; Wilks, A. R.; and Yan, C. H., <a href="http://arxiv.org/abs/math/0010298">Apollonian Circle Packings: Geometry and Group Theory I. The Apollonian Group.</a>, Discrete & Computational Geometry, 34 (2005), no. 4, 547-585.
%H A290508 Graham, R. L.; Lagarias, J. C.; Mallows, C. L.; Wilks, A. R.; and Yan, C. H., <a href="http://arxiv.org/abs/math/0010302">Apollonian Circle Packings: Geometry and Group Theory II. Super-Apollonian Group and Integral Packings.</a>, Discrete & Computational Geometry, 35 (2006), no. 1, 1-36.
%H A290508 Graham, R. L.; Lagarias, J. C.; Mallows, C. L.; Wilks, A. R.; and Yan, C. H., <a href="http://arxiv.org/abs/math/0010324">Apollonian Circle Packings: Geometry and Group Theory III. Higher Dimensions.</a>, Discrete & Computational Geometry, 35 (2006), no. 1, 37-72.
%H A290508 Graham, R. L.; Lagarias, J. C.; Mallows, C. L.; Wilks, A. R.; and Yan, C. H., <a href="http://arxiv.org/abs/math/0009113">Apollonian Circle Packings: Number Theory</a>, J. Number Theory, 100 (2003), 1-45.
%H A290508 N. J. A. Sloane, <a href="/A290508/a290508.png">Illustration for a(1)=12.</a> 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.
%H A290508 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SoddyCircles.html">Soddy Circles.</a>
%H A290508 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TangentCircles.html">Tangent Circles</a>
%H A290508 Wikipedia, <a href="https://en.wikipedia.org/wiki/Descartes%27_theorem">Descartes' theorem</a>
%e A290508 The table gives the first 8 examples:
%e A290508    a  b  c  d
%e A290508   == == == ==
%e A290508   12 12  3  1
%e A290508   15 15 10  2
%e A290508   24 24  2  1
%e A290508   63 56 56  9
%e A290508   69 46 23  6
%e A290508   70 30 21  5
%e A290508   72 45 45  8
%e A290508   80 80 36  9
%t A290508 aMax = 150; (* WARNING: O(n^3) *)
%t A290508 Do[
%t A290508      If[d // IntegerQ // Not, Continue[]];
%t A290508      If[GCD[a, b, c, d] > 1, Continue[]];
%t A290508      {a, b, c, d} // Sow;
%t A290508      , {a, aMax}, {b, a}, {c, b},
%t A290508        {d, {1/(1/a + 1/b + 1/c + 2 Sqrt[(a + b + c)/(a b c)])}}
%t A290508 ] // Reap // Last // Last // TableForm
%Y A290508 Cf. A290931 (4 mutually tangent circles, 1 circle enclosing 3).
%K A290508 nonn
%O A290508 1,1
%A A290508 _Albert Lau_, Aug 04 2017
%E A290508 Corrected (inserted 63, 72, 234, 275) and extended by _Ray Chandler_, Aug 06 2017
%E A290508 Edited by _N. J. A. Sloane_, Aug 16 2017