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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.

%I A386297 #27 Aug 01 2025 00:16:29
%S A386297 9,6,32,5,24,25,10,16,20,72,8,12,16,48,49,0,12,21,36,42,128,0,12,12,
%T A386297 28,30,112,81,0,13,12,24,28,60,54,200,0,10,16,12,24,62,48,140,121,0,
%U A386297 15,12,18,20,41,42,100,99,288,0,0,14,12,21,26,32,80,83,192,169
%N 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.
%C A386297 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).
%C A386297 T(n, k) = min(defect(S)) as S runs over all partitions of an n X n X n cuboid into k noncongruent cuboids.
%C A386297 A386296 gives the number of sets S.
%e A386297 Array begins
%e A386297    9     6     5     10
%e A386297   32    24    16     12
%e A386297   25    20    16     21
%e A386297   72    48    36     28
%e A386297   49    42    30     28
%e A386297  128    80    60     62
%e A386297   81    54    48     42
%e A386297  200   140   100     80
%e A386297 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.
%Y A386297 Cf. A081900, A385151, A385153, A385154, A386296.
%K A386297 tabl,nonn
%O A386297 3,1
%A A386297 _Janaka Rodrigo_, Jul 17 2025
%E A386297 More terms from _Sean A. Irvine_, Jul 29 2025