A128007 -id:A128007 - cp's OEIS frontend

A192493 Numerators of squared radii of circumcircles of non-degenerate triangles with integer vertex coordinates.

1, 1, 5, 25, 25, 2, 5, 25, 25, 13, 325, 169, 65, 4, 65, 17, 425, 221, 9, 289, 1105, 169, 85, 5, 325, 85, 50, 1105, 289, 25, 2125, 625, 13, 325, 425, 1625, 169, 1105, 125, 65, 29, 2465, 4225, 1885, 725, 377, 2465, 5525, 1885, 125, 8, 145, 65, 841, 17, 841, 845, 425, 2125, 221, 6409, 9425, 9, 325, 289, 145, 1105, 37, 5365, 3145, 169, 2405, 925, 85, 1369, 4625, 481, 625, 493, 2405, 10

Offset: 1

Hugo Pfoertner, Jul 10 2011

			The smallest triangle of lattice points {(0,0),(1,0),(0,1)} has circumradius R=sqrt(2)/2, i.e., R^2=1/2. Therefore a(1)=1, A192494(1)=2.

Hugo Pfoertner, Table of n, a(n) for n = 1..9089, covering range R^2 <= 100.
Hugo Pfoertner, Circles Passing through 3 Points of the Square Lattice, illustrations up to R^2=10.

Cf. A192494 (corresponding denominators), A128006, A128007.

Cf. A346993, A346994, A346995.

A128006 Numerators of rational-valued radii of circles given by 3 distinct integer points in the euclidean plane.

Original entry on oeis.org

1, 5, 5, 2, 17, 13, 5, 13, 29, 3, 37, 25, 13, 10, 17, 25, 15, 53, 65, 4, 65, 41, 29, 25, 17, 13, 41, 73, 65, 85, 5, 101, 61, 41, 26, 37, 85, 97, 65, 109, 61, 89, 325, 17, 125, 29, 53, 65, 6, 145, 85, 61, 37, 137, 25, 51, 13, 85, 205, 73, 173, 533, 20, 34, 89, 7, 197, 169, 113, 85

Offset: 1

Heinrich Ludwig, Feb 11 2007

A triangle in the Euclidean plane defines a circumcircle. There exist only certain rational-valued circumradii if the vertices of the triangle have integer coordinates. They are rendered by the pair of sequences A128006/A128007 in increasing order.

Illustration of A128006/A128007 using Plot 2.

See A128007 for denominators.

Cf. A128008, A128009, A128010, A128011, A192493, A192494.

A128008 Numerators of rational-valued radii of circles given by 3 distinct integer points in the 3-dimensional Euclidean space.

Original entry on oeis.org

1, 5, 3, 5, 11, 2, 17, 13, 9, 7, 33, 5, 13, 21, 27, 17, 29, 3, 37, 25, 19, 13, 33, 10, 17, 41, 7, 25, 29, 51, 11, 15, 53, 19, 99, 65, 23, 27, 4, 65, 57, 49, 41, 33, 29, 25, 21, 59, 17, 43, 69, 13, 53, 31, 9, 41, 73, 83, 37, 65, 14, 33, 85, 105, 43, 67, 77, 29, 63, 69, 89, 5

Offset: 1

Heinrich Ludwig, Feb 14 2007

A triangle in the 3-dimensional Euclidean space defines a circumcircle. There exist only certain rational-valued circumradii if the vertices of the triangle have integer coordinates. They are rendered by the pair of sequences A128008/A128009 in increasing order.

See A128009 for denominators. Cf. A128006, A128007, A128010, A128011.

A128010 Numerators of rational-valued radii of circles given by 3 distinct integer points in the 4-dimensional Euclidean space.

Original entry on oeis.org

1, 5, 3, 5, 7, 11, 2, 17, 13, 9, 7, 33, 19, 5, 13, 21, 27, 11, 14, 17, 29, 3, 37, 31, 25, 19, 13, 33, 10, 27, 17, 41, 7, 25, 18, 29, 51, 11, 37, 26, 15, 53, 19, 99, 65, 23, 27, 39, 47, 55, 91, 4, 65, 57, 49, 41, 33, 29, 25, 67, 21, 59, 17, 30, 43, 69, 13, 61, 35, 22, 53, 31, 9

Offset: 1

Heinrich Ludwig, Feb 18 2007

A triangle in the 4-dimensional Euclidean space defines a circumcircle. There exist only certain rational-valued circumradii if the vertices of the triangle have integer coordinates. They are listed by the pair of sequences A128010/A128011 in increasing order.

See A128011 for denominators. Cf. A128006, A128007, A128008, A128009.

A128009 Denominators of rational-valued radii of circles given by 3 distinct integer points in the 3-dimensional Euclidean space.

Original entry on oeis.org

1, 4, 2, 3, 6, 1, 8, 6, 4, 3, 14, 2, 5, 8, 10, 6, 10, 1, 12, 8, 6, 4, 10, 3, 5, 12, 2, 7, 8, 14, 3, 4, 14, 5, 26, 17, 6, 7, 1, 16, 14, 12, 10, 8, 7, 6, 5, 14, 4, 10, 16, 3, 12, 7, 2, 9, 16, 18, 8, 14, 3, 7, 18, 22, 9, 14, 16, 6, 13, 14, 18, 1

Offset: 1

Heinrich Ludwig, Feb 14 2007

See A128008.

See A128008 for numerators. Cf. A128006, A128007, A128010, A128011.

A128011 Denominators of rational-valued radii of circles given by 3 distinct integer points in the 4-dimensional Euclidean space.

Original entry on oeis.org

1, 4, 2, 3, 4, 6, 1, 8, 6, 4, 3, 14, 8, 2, 5, 8, 10, 4, 5, 6, 10, 1, 12, 10, 8, 6, 4, 10, 3, 8, 5, 12, 2, 7, 5, 8, 14, 3, 10, 7, 4, 14, 5, 26, 17, 6, 7, 10, 12, 14, 23, 1, 16, 14, 12, 10, 8, 7, 6, 16, 5, 14, 4, 7, 10, 16, 3, 14, 8, 5, 12, 7, 2

Offset: 1

Heinrich Ludwig, Feb 18 2007

See A128010 for numerators and further information. Cf. A128006, A128007, A128008, A128009.

A331244 Triangles with integer sides i <= j <= k sorted by radius of enclosing circle, and, in case of ties, lexicographically by side lengths (smallest first). The sequence gives the shortest side i. The other sides are in A331245 and A331246.

Original entry on oeis.org

1, 1, 2, 2, 1, 2, 3, 2, 3, 1, 2, 3, 4, 2, 3, 3, 1, 2, 4, 3, 4, 5, 2, 3, 3, 4, 1, 4, 2, 3, 5, 4, 5, 6, 2, 3, 3, 4, 4, 5, 4, 1, 2, 5, 3, 4, 6, 5, 6, 2, 3, 3, 4, 4, 5, 5, 4, 1, 6, 2, 5, 7, 3, 4, 6, 5, 7, 6, 7, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 1, 5, 2, 3, 7, 6, 4

Offset: 1

Hugo Pfoertner, Jan 20 2020

nonn

The enclosing circle differs from the circumcircle by limiting the radius to (longest side)/2 for obtuse triangles, i.e., those with i^2 + j^2 < k^2.

			List of triangles begins:
   n
   |     R^2
   |     |    i .... (this sequence)
   |     |    | j .. (A331245)
   |     |    | | k  (A331246)
   |     |    | | |
   1    1/ 3  1 1 1
   2   16/15  1 2 2
   3    4/ 3  2 2 2
   4    9/ 4  2 2 3  obtuse
   5   81/35  1 3 3
   6   81/32  2 3 3
   7    3/ 1  3 3 3
   8    4/ 1  2 3 4  obtuse
   9   81/20  3 3 4
  10  256/63  1 4 4
  11   64/15  2 4 4
  12  256/55  3 4 4
  13   16/ 3  4 4 4
  14   25/ 4  2 4 5  obtuse
  15   25/ 4  3 3 5  obtuse
  16   25/ 4  3 4 5
  17  625/99  1 5 5

Cf. A128006, A128007, A192493, A192494, A331227, A331228, A331251, A331252, A331253, A331254, A331255, A331256.

Cf. A331245 (middle side), A331246 (longest side).

A387045 Positive numbers k with property that the largest circle on the xy-plane enclosing exactly k lattice points in its interior does not exist.

Original entry on oeis.org

5, 6, 17, 18, 33, 34, 35, 36, 38, 50, 53, 54, 63, 70, 71, 72, 73, 89, 90, 97, 98, 102, 109, 110, 125, 126, 127, 128, 129, 150, 151, 165, 167, 168, 178, 188, 198, 209, 210, 217, 218, 219, 220, 221, 222, 242, 243, 257, 258, 259, 260, 277, 278, 285, 286, 294

Offset: 1

Jianqiang Zhao, Aug 14 2025

Conjecture: This sequence is infinite.

			It can be proved that the largest circle enclosing exactly 5 or 6 lattice points in the interior on the xy-plane does not exist. Number 5 is the smallest nonnegative integer having this property and 6 is the next. Therefore, a(1)=5 and a(2)=6.
Here is a brief argument. For details, please see my arxiv paper 2505.06234.
First, let C be the circle going through (-1,0) centered at (1/2,1/2). It passes exactly 8 lattice points and encloses exactly 4. Now with (-1,0) fixed on the circle we can shrink C by an infinitesimal amount to circle C' so that C' only goes through one lattice point (-1,0). Then another infinitesimal perturbation will move C' to include exactly 5 lattice points in its interior. Another infinitesimal perturbation will move C' to include exactly 6 lattice points in its interior. Therefore, if the largest circle enclosing exactly 5 or 6 interior lattice points exists, then its radius is at least sqrt(10)/2.
Second, a geometric argument shows that if the radius of a circle is at least sqrt(10)/2 then it encloses either exactly 4 interior lattice points or at least 7 interior lattice points.

Jianqiang Zhao, Table of n, a(n) for n = 1..189
Jianqiang Zhao, The Largest Circle Enclosing n Lattice Points, arXiv:2505.06234 [math.GM], 2025. This is an expanded version of the paper by Jianqiang Zhao, The Largest Circle Enclosing n Lattice Points, Geometry, Vol. 2 (2025), 12.

Cf. A387044, complement of A387045; A192493, A192494, A128006, A128007.

cp's OEIS Frontend

A192493 Numerators of squared radii of circumcircles of non-degenerate triangles with integer vertex coordinates.

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A128006 Numerators of rational-valued radii of circles given by 3 distinct integer points in the euclidean plane.

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A128008 Numerators of rational-valued radii of circles given by 3 distinct integer points in the 3-dimensional Euclidean space.

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A128010 Numerators of rational-valued radii of circles given by 3 distinct integer points in the 4-dimensional Euclidean space.

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A128009 Denominators of rational-valued radii of circles given by 3 distinct integer points in the 3-dimensional Euclidean space.

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A128011 Denominators of rational-valued radii of circles given by 3 distinct integer points in the 4-dimensional Euclidean space.

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A331244 Triangles with integer sides i <= j <= k sorted by radius of enclosing circle, and, in case of ties, lexicographically by side lengths (smallest first). The sequence gives the shortest side i. The other sides are in A331245 and A331246.

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A387045 Positive numbers k with property that the largest circle on the xy-plane enclosing exactly k lattice points in its interior does not exist.

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