A386297 Array read by antidiagonals T(n,k) is the minimal defect across all partitions of an n X n X n cube into k noncongruent cuboids, or 0 if there is no such partition.

9, 6, 32, 5, 24, 25, 10, 16, 20, 72, 8, 12, 16, 48, 49, 0, 12, 21, 36, 42, 128, 0, 12, 12, 28, 30, 112, 81, 0, 13, 12, 24, 28, 60, 54, 200, 0, 10, 16, 12, 24, 62, 48, 140, 121, 0, 15, 12, 18, 20, 41, 42, 100, 99, 288, 0, 0, 14, 12, 21, 26, 32, 80, 83, 192, 169

Offset: 3

Author

Janaka Rodrigo, Jul 17 2025

Keywords

• tabl
• nonn

Comments

Let V(x,y,z)=x*y*z be the volume of a cuboid (x,y,z). For a given set of cuboids 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 = max(S)-min(S).

T(n, k) = min(defect(S)) as S runs over all partitions of an n X n X n cuboid into k noncongruent cuboids.

A386296 gives the number of sets S.

Examples

			Array begins
   9     6     5     10
  32    24    16     12
  25    20    16     21
  72    48    36     28
  49    42    30     28
 128    80    60     62
  81    54    48     42
 200   140   100     80
The only set S of distinct six cuboids filling 3 X 3 X 3 cube in triplet form is, S = {(1,1,1), (1,1,2), (1,1,3), (1,2,2), (2,2,2), (1,3,3)} giving Min(S)=1, Max(S)=9, and defect(S) = 9-1 = 8. Since this is the only defect T(3,6)=8.
		

Crossrefs

Cf. A081900, A385151, A385153, A385154, A386296.

Extensions

More terms from Sean A. Irvine, Jul 29 2025