A239671 Magic constants of the magic cubes 3 X 3 X 3 composed of prime numbers.

3309, 4659, 5091, 5433, 7179, 7431, 7773, 7863, 8223, 8367, 8403, 9501, 9543, 9573, 9987, 10029, 10113, 10371, 10551, 10821

Offset: 1

Author

Natalia Makarova, Mar 23 2014

Keywords

• nonn

Comments

A magic cube is the 3-dimensional equivalent of a magic square, that is, n^3 distinct integers arranged in an n X n X n grid such that the sum of the integers in each row, each column, each pillar, and the four main space diagonals is equal to the same number, called magic constant of the cube.

The magic cube is associative if the sum of any 2 numbers, symmetrically located relative to the center of the cube, is equal to the same number, called constant of associativity of the cube.

Magic cubes of order 3 are simple magic cubes.

All magic cubes of order 3 are associative.

The first two prime magic cubes of order 3 were found by Akio Suzuki in 1977 (see Prime Number Magic Cubes link).

The general formula of the magic cube of order 3:

......................................................

. x1, x2, 3k/2-x1-x2,

. x3, x4, 3k/2-x3-x4,

. 3k/2-x1-x3, 3k/2-x2-x4, -3k/2+x1+x2+x3+x4,

.......................................................

. -k+x2+x3+x4, 2k-2*x2-x4, k/2+x2-x3,

. 2k-2*x3-x4, k/2, -k+2*x3+x4,

. k/2-x2+x3, -k+2*x2+x4, 2k-x2-x3-x4,

.......................................................

. 5k/2-x1-x2-x3-x4, -k/2+x2+x4, -k/2+x1+x3,

. -k/2+x3+x4, k-x4, k-x3,

. -k/2+x1+x2, k-x2, k-x1

........................................................

Here k is the constant of associativity (any even number), x1, x2, x3, x4 are any integers.

Examples

			For n = 3, a(3) = 5091.
......................
.  1061  3167   863
.  2243   431  2417
.  1787  1493  1811
......................
.  2447    23  2621
.  1871  1697  1523
.   773  3371   947
......................
.  1583  1901  1607
.   977  2963  1151
.  2531   227  2333
......................
		

Links

• Harvey Heinz, Prime Number Magic Cubes
• Natalia Makarova, Discussion on scientific forum (in Russian)
• Wikipedia, Magic cube