Magic Constant

The number

			(1)
			(2)

to which the numbers in any horizontal, vertical, or main diagonal line must sum in a magic square. The first few values are 1, 5, 15, 34, 65, 111, 175, 260, ... (OEIS A006003). The magic constant for an th order magic square starting with an integer and with entries in an increasing arithmetic series with difference between terms is

(3)

(Hunter and Madachy 1975, Madachy 1979). In a panmagic square, in addition to the main diagonals, the broken diagonals also sum to .

For a magic cube, magic tesseract, etc., the magic -D constant is

			(4)
			(5)

The first few magic constants are summarized in the following table.

Sloane	A006003	A027441	A021003
1	1	1	1
2	5	9	17
3	15	42	123
4	34	130	514
5	65	315	1565

There is a corresponding multiplicative magic constant for multiplication magic squares.

A similar magic constant of degree is defined for magic series and multimagic series as times the sum of the first th powers,

			(6)
			(7)

where is a harmonic number of order . The following table gives the first few values.

Sloane	A006003	A052459	A052460	A052461
1	1	1	1	1
2	5	15	50	177
3	15	95	675	5111
4	34	374	4624	60962
5	65	1105	21125	430729