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Similar to the Mondrian Art sequence (A276523), but allowing repetition of rectangles with different orientations.

Proved optimal to a(45) by R. Gerbicz. Best values known for a(46)-a(96): 10, 12, 11, 12, 12, 8, 12, 12, 13, 12, 12, 14, 14, 15, 12, 15, 14, 15, 14, 16, 16, 15, 16, 16, 16, 17, 16, 17, 14, 17, 18, 16, 18, 16, 18, 15, 16, 18, 18, 16, 18, 17, 19, 20, 17, 17, 21, 20, 20, 21, 22.

Seems to be bounded above by ceiling(n/log(n)). The currently verified distances from this bound are 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 3, 1, 2, 2, 2, 2, 2, 2 (A279848).

If ceiling(n/log(n)) + 3 is an upper bound for the Mondrian Art Problem (A276523), a(n) is the amount by which the optimal value beats the upper bound.

Terms a(86) and a(139) are at least 5. Term a(280) is at least 7.

Term a(138) is at least 9, defect 22 (1200-1178) with 16 rectangles.

Best values known for a(66) to a(96): 3, 1, 1, 2, 1, 2, 0, 1, 1, 1, 2, 1, 0, 1, 2, 1, 2, 2, 1, 1, 5, 1, 3, 1, 0, 1, 2, 2, 0, 0, 1.

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

Alternatively, a(n) is the optimal solution when an n X n X n cube is partitioning into 3 cuboids of different dimensions.

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

Let V(x,y,z) = x*y*z be the volume and for a given set of triplets S, 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 least possible value of the defect as S runs over the possible partitions of the n X n X n cuboid into 3 cuboids of different dimensions.

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.

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

Alternatively, a(n) is the optimal solution when an n X n X n cube is partitioning into five cuboids of different dimensions.

Let elements of the unordered integer triplet (x,y,z) be the dimensions of a cuboid in a set of five cuboids and volume V(x,y,z) = x*y*z; cuboids have five values for each set of five triplets S, produced by the union of A(n), B(n), C(n), where A(n), B(n), and C(n) are sequences of sets as introduced in A384479.

Define min(S) = min{V(x,y,z):(x,y,z) in S} and max(S) = max{V(x,y,z):(x,y,z) in S}, then defect(S) = max(S) - min(S).

a(n) is the smallest possible value of defect(S) where S runs over all possible ways of partitioning the n X n X n cube into five cuboids of different dimensions.

If ceiling(n/log(n)) is an upper bound for the Mondrian Art Problem variant (A279596), a(n) is the amount by which the optimal value beats the upper bound.

Terms a(3) to a(45) verified optimal by R. Gerbicz.

Term a(103) is at least 9, defect 14 (630-616) with 17 rectangles.

Best values known for a(46) to a(96): 3, 1, 2, 1, 1, 5, 2, 2, 1, 2, 2, 1, 1, 0, 3, 0, 2, 1, 2, 0, 0, 1, 1, 1, 1, 0, 1, 1, 4, 1, 0, 2, 0, 3, 1, 4, 3, 1, 1, 4, 2, 3, 1, 0, 4, 4, 0, 1, 1, 0, 0.