Most-perfect magic squares

A magic square is said to be most-perfect, if it contains the consecutive numbers 1, 2, …, n2 such that

  1. any four adjacent integers forming a 2x2-subsquare sum to

    Supermagisch-Formel

  2. any pair of integers distant n/2 along a diagonal sum to

    T=n2 + 1

  3. it is a double-even magic square, i.e. of order n=4k.

The following magic square with order 4 and 8 are most-perfect.

115108
126313
79162
144511
116173253603744
635047341162722
314193055583942
61524536982524
125282164494833
545938432151831
107262362514635
565740414132029