a(3) = 3 because the 3 X 3 X 3 cube can be divided into subcubes in 3 different ways: a single 3 X 3 X 3 cube, a 2 X 2 X 2 plus 19 1 X 1 X 1 cubes, or 27 1 X 1 X 1 cubes.
a(4) = 11 because the 4 X 4 X 4 cube can be divided into 11 different combinations of subcubes. The table below lists each of the 11 combinations and gives the number of ways those subcubes can be arranged:
(1) 64 1 X 1 X 1 cubes in 1 way
(2) 56 1 X 1 X 1 cubes and 1 2 X 2 X 2 cube in 27 ways
(3) 48 1 X 1 X 1 cubes and 2 2 X 2 X 2 cubes in 193 ways
(4) 40 1 X 1 X 1 cubes and 3 2 X 2 X 2 cubes in 544 ways
(5) 32 1 X 1 X 1 cubes and 4 2 X 2 X 2 cubes in 707 ways
(6) 24 1 X 1 X 1 cubes and 5 2 X 2 X 2 cubes in 454 ways
(7) 16 1 X 1 X 1 cubes and 6 2 X 2 X 2 cubes in 142 ways
(8) 8 1 X 1 X 1 cubes and 7 2 X 2 X 2 cubes in 20 ways
(9) 8 2 X 2 X 2 cubes in 1 way
(10) 37 1 X 1 X 1 cubes and 1 3 X 3 X 3 cube in 8 ways
(11) 1 4 X 4 X 4 cube in 1 way
The total number of arrangements is 2098 = A228267(4,4,4).
