A385153 a(n) is the least possible difference between the largest and smallest volumes of distinct four-cuboid combinations filling an n X n X n cube.

5, 16, 16, 36, 30, 60, 48, 100, 83, 96, 123, 182, 130, 264, 182, 324, 224, 280, 259, 484, 369, 576, 449, 676, 423, 560, 528, 900, 598, 1008, 638, 1054, 859, 864, 979, 1330, 884, 1620, 1054, 1764, 1152, 1364, 1185, 2116, 1553, 2304, 1713, 2500, 1513, 1924, 1760

Offset: 3

Author

Janaka Rodrigo, Jun 19 2025

Keywords

• nonn

Comments

Developed as the three-dimensional extension of the Mondrian Art Problem.

Alternatively, a(n) is the minimum defect when an n X n X n cube is partitioning into four cuboids of different dimensions.

Let elements of the unordered integer triplet (x,y,z) be the dimensions of a cuboid in a set S of four cuboids.

Let V(x,y,z) = x*y*z be the volume and for a given element of S. Define min(S) = min{V(x,y,z): (x,y,z) in S}, max(S) = max{V(x,y,z): (x,y,z) in S}, and defect(S) = max(S)-min(S).

a(n) is the smallest value of the defect(S) across all possible partitions of the n X n X n cuboid into four cuboids of different dimensions.

Examples

			The 4 X 4 X 4 cube has 12 different ways of partitioning into four distinct cuboids each giving a defect as the difference between the largest volume and the smallest volume relevant to the four cuboids of the set.
The optimal solution is given by the set {(4,2,1), (4,2,2), (4,3,2), (4,4,1)} because it has the minimum defect.
The least possible defect = max(8,16,24,16)-min(8,16,24,16) = 24-8 = 16.
Therefore, a(4) = 16.
		

Links

• Janaka Rodrigo, Python program

Crossrefs

Cf. A276523, A384311.

Extensions

More terms from Sean A. Irvine, Jul 16 2025