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A cousin prime pair is a pair of primes that differ by 4.

a(8) was found by Petukhov.

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

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.

The magic constants of most-perfect magic squares of order 4 composed of distinct primes see A191533.

The minimal magic constant of most-perfect magic square of order 6 composed of distinct primes corresponds to a(1) = 29790, see A258082.

The seven terms shown have been verified by exhaustive search. - Natalia Makarova, Jun 09 2016

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

All most-perfect magic squares are pandiagonal.

All pandiagonal magic squares of order 4 are most-perfect (cf. A191533).

A magic square is associative if the sum of any two elements symmetric about its center is the same. A magic square is pandiagonal if the sum of the numbers in any broken diagonal equals the magic constant. A magic square is ultramagic if it is associative and pandiagonal.

Ultramagic squares exist for orders n>=5.

The following bounds for the next terms are known: 12249<=a(9)<=13059, 4200<=a(10)<=46150, a(11)>=26521, a(12)>=8820, a(13)>=49439, a(14)>=16170, a(15)>=74595, a(16)>=21840.

A magic cube is associative if the sum of its any two elements that are symmetric about the cube center equals the same number, called the associativity constant of the cube.

All magic 3 X 3 X 3 cubes are associative.

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.

Stanley antimagic square of index d and order n is an n X n matrix where the sum of any n matrix elements in pairwise distinct rows and pairwise distinct columns equals d.