A390583 - OEIS

%I #10 Nov 17 2025 21:13:31

%S 1,2,24,24,1,3,12,12,2,3,120,120,2,10,15,15,3,4,336,336,3,5,120,120,3,

%T 8,22,264,4,5,720,720,4,11,33,132,5,6,1320,1320,5,7,420,420,5,8,117,

%U 4680,5,21,30,70,6,7,2184,2184,6,14,105,105,6,19,32,1824

%N Rectangular array read by rows of ordered quadruples of coprime positive integers {r1..r4} for which circles of radius r1..r4 can be arranged as mutually externally tangent to one another, with rows in lexicographic order.

%C Primitive positive integer solutions to Descartes' theorem for 4 kissing circles (radii r1..r4): (1/r1 + 1/r2 + 1/r3 + 1/r4) ^ 2 = 2 * ((1/r1)^2 + (1/r2)^2 + (1/r3)^2 + (1/r4)^2).

%C There are only a finite number of rows for each distinct r1 because r2 must be <= r1/(2/sqrt(3)-1) (otherwise r3 or r4 would be < r2), r3 must be <= 4*(1/r1+1/r2)/(1/r1-1/r2)^2 (otherwise r4 would be < r3), and there exists only 0 or 1 possible r4 given a known r1..r3.

%C It is observed and generally conjectured that if the largest 2 terms of a quadruple are equal, then the absolute difference between the other 2 terms is 1 or 2 times a cube.

%C Regular quadruples include {k, k+1, A069074(k-1), A069074(k-1)} for every positive k, {k, k+2, A000447(k)*12, A000447(k)*12} for every odd positive k, and {k, k+8, A061550((k-2)/4), A061550((k-2)/4)} for every positive k = 2 mod 4.

%C Some pairs of quadruples contain coincident triples; the smallest are {2, 3, 120, 120} and {3, 5, 120, 120}. It is unknown whether arbitrarily large chains of such pairs can exist (e.g. {a, b, c, d} -> {b, c, d, e} -> {c, d, e, f} -> etc.).

%H Charles L. Hohn, <a href="/A390583/b390583.txt">Table of n, a(n) for n = 1..4000</a>

%e Quadruples of circle radii r1..r4:

%e   r1 r2   r3   r4

%e    1  2   24   24

%e    1  3   12   12

%e    2  3  120  120

%e    2 10   15   15

%e    3  4  336  336

%e    3  5  120  120

%e    3  8   22  264

%e    4  5  720  720

%e    4 11   33  132

%e    5  6 1320 1320

%e   ...

%o (PARI) my(r1, r2, r3, r4, a, b, c, t); for(r1=1, 100, for(r2=r1+1, ceil(r1/(2/sqrt(3)-1)), a=1/r1; b=1/r2; for(r3=r2, floor(4*(a+b)/(a-b)^2), a=r2*r3; b=r1*r3; c=r1*r2; t=a*b+a*c+b*c; if(!issquare(t), next); t=a+b+c-2*sqrtint(t); if(t<=0, next); r4=r1*r2*r3/t; if(type(r4)!="t_INT" || gcd([r1, r2, r3, r4])>1, next); print([r1, r2, r3, r4]))))

%Y Equivalent for 4 spheres and a plane: A390148.

%Y Column 4, independently sorted: A290508.

%Y Cf. A069074, A000447, A061550.

%K nonn,tabf

%O 1,2

%A _Charles L. Hohn_, Nov 11 2025