A046160 Bends of spheres in Soddy's bowl of integers which are tangent to the unit sphere and a single 2-sphere.

2, 5, 6, 9, 11, 14, 15, 18, 21, 23, 27, 29, 30, 33, 38, 39, 41, 45, 50, 51, 54, 59, 63, 65, 66, 69, 75, 77, 78, 81, 83, 86, 93, 95, 99, 102, 105, 110, 111, 113, 114, 119, 123, 126, 129, 131, 135, 141, 146, 149, 150, 153, 158, 159, 165, 171, 173, 174, 177, 183

Offset

1,1

Comments

From Martin Fuller, Mar 28 2026: (Start)

Apart from the first term, the definition corresponds to Fig 2 in Soddy's paper, and to this text opposite the figure: "successive rings I, 3 ; II, 6, 5 ; III, 11, 9, 9 ; IV, 18, 15, 14, 15 ; V, 27, 23, 21, 21, 23 ; VI, 38, 33, 30, 29, 30, 33 and so on."

The list of all bends in the bowl is given by A045506. The smallest number in A045506 not in A046159 or A046160 is 8; this sphere is not shown in Fig 2 but would be tangent to (5,3,3,-1), outside the displayed spheres, each side of the equator.

Conjecture: the list of bends of all spheres not in the plane is A049636 "Congruent to 0 or 2 mod 3, but not equal to 0 or 3." (End)

Links

Table of n, a(n) for n=1..60.

Frederick Soddy, The Bowl of Integers and the Hexlet, Nature 139, 77-79, 1937 [Accessible copy]

Eric Weisstein's World of Mathematics, Bowl of Integers.

Example

a(1) = 2 is the bend of the two "initial" spheres of radius 1/2, centered in (0, 0, +- 0.5), touching each other at the origin, and the unit sphere in (0, 0, +-1). Next there are six spheres of radius 1/3, centered in the plane, each of which touches its two neighbor spheres with bend 3, and also the two spheres of bend 2, and also the outer sphere of bend -1. Since their center is on the plane, their bend is not listed here.
a(2) = 5 is the bend of the two times six spheres that can be filled in between the spheres of bend 2 and the above mentioned spheres of bend 3, six above and six below the plane z=0. Since their center is not in the plane, their bend is again listed here.

Prog

(PARI) a(limit)={
  my(L = List([[5, 3, 2, -1]]), R = List(2));
  while(1,
    if(#R >= limit, return(Vec(R)));
    listsort(~L);
    my(k = L[1]);
    listpop(~L, 1);
    if(k[1] > R[#R], listput(~R, k[1]));
    my(s = vecsum(k), ss = norml2(k));
    my(kn = (sqrtint((#k-1)*(s^2 - (#k-2)*ss)) + s) / (#k-2));
    if(kn > k[1], listput(~L, [kn, k[2], k[3], k[4]]));
    if(kn >= k[1] && kn > k[2], listput(~L, [kn, k[1], k[3], k[4]]));
  );
}; \\ Martin Fuller, Mar 28 2026

Crossrefs

Cf. A046159 (bends of spheres centered on the plane).

Cf. A045506 (list of all bends: -1, 2, 3, 5, ...).

Sequence in context: A244737 A285347 A187836 * A033161 A024516 A286341

Adjacent sequences: A046157 A046158 A046159 * A046161 A046162 A046163

Keyword

nonn,nice

Author

Eric W. Weisstein

Extensions

Description revised, a(12)=29 inserted and a(15) onward from Martin Fuller, Mar 28 2026

Status

approved