A259733 The magic constants of most-perfect magic squares of order 8 composed of distinct prime numbers.

24024, 26040, 43680, 44352, 44520, 44880

Offset

1,1

Comments

A magic square of order n = 2k is most-perfect if the following two conditions hold: (i) every 2 X 2 subsquare (including wrap-around) sums to 2T; and (ii) any pair of elements at distance k along a diagonal or a skew diagonal sums to T, where T = S/k, S is the magic constant.

All most-perfect magic squares are pandiagonal.

All pandiagonal magic squares of order 4 are most-perfect, see A191533.

The magic constants of most-perfect magic squares of order 6 composed of distinct primes see A258755.

The minimal magic constant of most-perfect magic square of order 8 composed of distinct primes corresponds to a(1) = 24024, see A258082.

It seems that only the first term, or possibly the first two terms, have been proved to be correct. The other terms are conjectural (that is, there may be missing terms). - N. J. A. Sloane, Jul 28 2015

Links

Table of n, a(n) for n=1..6.

N. Makarova and others, Magic squares, discussion at the scientific forum dxdy.ru (in Russian), Feb. 2015.

N. Makarova, Puzzle 671: Most Perfect Magic Squares, Prime Puzzles & Problems.

Natalia Makarova, Most-perfect magic squares of order 8

Wikipedia, Most-perfect magic square

Example

a(2) = 26040 corresponds to the following most-perfect magic square by N. Makarova:
    61 6229  661 5563 2087 4643 1487 5309
  3719 3011 3119 3677 1693 4597 2293 3931
  1777 4513 2377 3847 3803 2927 3203 3593
  4139 2591 3539 3257 2113 4177 2713 3511
  4423 1867 5023 1201 6449  281 5849  947
  4817 1913 4217 2579 2791 3499 3391 2833
  2707 3583 3307 2917 4733 1997 4133 2663
  4397 2333 3797 2999 2371 3919 2971 3253
a(3) = 43680 corresponds to the following most-perfect magic square by S. Zorkin:
    229 10457  859 9767  7393  3761  6763 4451
   7841  3313 7211 4003   677 10009  1307 9319
    953  9733 1583 9043  8117  3037  7487 3727
   8623  2531 7993 3221  1459  9227  2089 8537
   3527  7159 4157 6469 10691   463 10061 1153
  10243   911 9613 1601  3079  7607  3709 6917
   2803  7883 3433 7193  9967  1187  9337 1877
   9461  1693 8831 2383  2297  8389  2927 7699

Crossrefs

Cf. A191533, A258082, A258755.

Sequence in context: A233882 A249625 A263036 * A295451 A084691 A084692

Adjacent sequences: A259730 A259731 A259732 * A259734 A259735 A259736

Keyword

nonn,more

Author

Natalia Makarova and Sergey Zorkin, Jul 04 2015

Status

approved