A191533 Magic constants of pandiagonal magic squares of order 4 composed of distinct primes.

240, 252, 288, 372, 408, 420, 480, 492, 504, 528, 540, 552, 560, 564, 576, 588, 600, 612, 620, 624, 648, 660, 672, 680, 684, 708, 720, 728, 732, 740, 756, 768, 780, 792, 800, 816, 828, 836, 840, 848, 852, 860, 864, 876, 888, 900, 912, 920, 924, 936, 948

Offset: 1

Author

Natalia Makarova, Jun 05 2011

Keywords

• nonn

Comments

A pandiagonal square of order 4 consists of 8 pairs of complementary numbers with the sum in each pair equal to S/2 (where S is the magic constant). For example, the array {7, 113, 11, 109, 13, 107, 17, 103, 19, 101, 23, 97, 31, 89, 37, 83, 41, 79, 47, 73, 53, 67, 59, 61} consists of 12 complementary prime pairs with the sum 7 + 113 = 11 + 109 = ... = 59 + 61 = 120 = S/2.

Pandiagonal squares of order 4 are also the most-perfect squares.

There is a one-to-one correspondence between pandiagonal and associative magic squares of order 4. Any pandiagonal square can be turned into an associative square by rearrangements of its rows and columns, and vice versa.

For example, pandiagonal square:

[ 13 83 31 113

97 47 79 17

89 7 107 37

41 103 23 73 ]

the corresponding associative square:

[ 13 83 113 31

97 47 17 79

41 103 73 23

89 7 37 107]

Magic constants of pandiagonal magic squares of order 4 are always multiples of 4. It looks as though most sufficiently large multiples of 4 are magic constants of some pandiagonal magic squares of order 4. For multiples of 4 between 3000 and 10000, only 3028, 3208, 3436, 3664, 4436, 4504, and 5116 are not the magic constant of any pandiagonal magic squares of order 4. - Zhao Hui Du, Jan 09 2024

Examples

			a(3)=288 for the matrix
  [ 7 127  41 113
   71  83  37  97
  103  31 137  17
  107  47  73  61 ]
		

Links

• Max Alekseyev, Table of n, a(n) for n = 1..100
• Natalia Makarova, Order-4 pandiagonal magic squares composed of primes (in Russian)

Crossrefs

Cf. A179440.

Extensions

Terms a(18) onward from Max Alekseyev, May 26 2012