cp's OEIS Frontend

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.

%I A385153 #30 Jul 16 2025 23:35:05
%S A385153 5,16,16,36,30,60,48,100,83,96,123,182,130,264,182,324,224,280,259,
%T A385153 484,369,576,449,676,423,560,528,900,598,1008,638,1054,859,864,979,
%U A385153 1330,884,1620,1054,1764,1152,1364,1185,2116,1553,2304,1713,2500,1513,1924,1760
%N 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.
%C A385153 Developed as the three-dimensional extension of the Mondrian Art Problem.
%C A385153 Alternatively, a(n) is the minimum defect when an n X n X n cube is partitioning into four cuboids of different dimensions.
%C A385153 Let elements of the unordered integer triplet (x,y,z) be the dimensions of a cuboid in a set S of four cuboids.
%C A385153 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).
%C A385153 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.
%H A385153 Janaka Rodrigo, <a href="/A385153/a385153.txt">Python program</a>
%e A385153 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.
%e A385153 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.
%e A385153 The least possible defect = max(8,16,24,16)-min(8,16,24,16) = 24-8 = 16.
%e A385153 Therefore, a(4) = 16.
%Y A385153 Cf. A276523, A384311.
%K A385153 nonn
%O A385153 3,1
%A A385153 _Janaka Rodrigo_, Jun 19 2025
%E A385153 More terms from _Sean A. Irvine_, Jul 16 2025