A387044 Nonnegative numbers k for which there exists a largest circle enclosing exactly k points (strictly inside) on a square grid, as opposed to only a least upper bound on that circle size.

0, 1, 2, 3, 4, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 37, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 74, 75, 76, 77, 78, 79, 80, 81, 82

Offset

1,3

Comments

Steinhaus shows that a circle enclosing exactly k points always exists.

The question for k is whether the set of circle sizes enclosing exactly k points has a maximum, or only a least upper bound.

This sequence is unbounded.

References

H. Steinhaus, One Hundred Problems in Elementary Mathematics, Basic Book Inc. Publisher, New York, 1964.

Links

Jianqiang Zhao, Table of n, a(n) for n = 1..911

Jianqiang Zhao, The Largest Circle Enclosing n Lattice Points, arXiv:2505.06234 [math.GM], 2025, being the "maximally circlable" numbers.

Jianqiang Zhao, The Largest Circle Enclosing n Lattice Points, Geometry, Vol. 2 (2025), 12.

Example

4 is a term since a circle of radius sqrt(10)/2 encloses exactly 4 points, and no larger circle does so.
5 is not a term since there are circles strictly smaller than radius sqrt(10)/2 but arbitrarily close to it which enclose exactly 5 points; but at sqrt(10)/2 or greater they cannot, so there is no largest circle, only a least upper bound (sqrt(10)/2).

Crossrefs

Cf. A387045 (complement); A192493, A192494, A128006, A128007.

Sequence in context: A109436 A177045 A241783 * A039254 A039195 A351173

Adjacent sequences: A387041 A387042 A387043 * A387045 A387046 A387047

Keyword

nonn

Author

Jianqiang Zhao, Aug 14 2025

Status

approved