A257316 Smallest magic constant of ultramagic squares of order n composed of distinct prime numbers.

3505, 990, 4613, 2040

Offset

5,1

Comments

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.

Links

Table of n, a(n) for n=5..8.

Discussion at the scientific forum dxdy.ru, Devilish magic squares of primes (in Russian)

Wikipedia, Magic Square

Example

a(6)=990 corresponds to the following ultramagic square found by Max Alekseyev:
  103  59 163 233 139 293
  229 257 307 131  13  53
  283  17  67 173 181 269
   61 149 157 263 313  47
  277 317 199  23  73 101
   37 191  97 167 271 227
a(7)=4613 corresponds to the following ultramagic square found by Natalia Makarova:
   227  617  677  431 1217 1307  137
  1259  827 1061  509  521  167  269
   347  929 1187   17  557  719  857
    89  479   29  659 1289  839 1229
   461  599  761 1301  131  389  971
  1049 1151  797  809  257  491   59
  1181   11  101  887  641  701 1091
a(8)=2040 corresponds to the following ultramagic square found by Natalia Makarova:
  241 199 409 467  47  79 359 239
  421 137   7  53 487 179 317 439
   31 281 347 353 227 277 127 397
  449 197 109 379 491 337  11  67
  443 499 173  19 131 401 313  61
  113 383 233 283 157 163 229 479
   71 193 331  23 457 503 373  89
  271 151 431 463  43 101 311 269

Crossrefs

Cf. A006052, A081262, A081263.

Sequence in context: A108117 A233992 A203845 * A224724 A338978 A274237

Adjacent sequences: A257313 A257314 A257315 * A257317 A257318 A257319

Keyword

nonn,more

Author

Natalia Makarova, Apr 20 2015

Status

approved