A164843 The smallest magic constant of an n X n magic square with distinct prime entries.

177, 120, 233, 432, 733, 1154, 1731, 2470, 3417, 4584, 6013, 7712, 9731, 12088, 14807, 17940, 21501, 25530, 30021, 35086, 40675, 46840, 53631, 61092, 69251, 78100, 87697, 98084, 109309, 121380, 134377, 148258, 163043

Offset

3,1

Comments

a(n) >= m(n), where m(n) is the smallest integer of the same parity as n, which is >= (Sum_{k=1..n^2} prime(k+1))/n. For example, Sum_{k=1..5^2} prime(k+1)/5=231.8, so m(5)=233. Conjecture: for n > 4, a(n)=m(n) or a(n)=m(n)+2.

Links

Table of n, a(n) for n=3..35.

Harvey Heinz, Prime magic squares

A. Lelechenko and N. Makarova, Examples of prime magic n X n squares with minimal magic constant for n=5..13.

N. Makarova, Smallest prime magic squares, Part I (in Russian)

N. Makarova, Smallest prime magic squares, Part II (in Russian)

Mathworld, Prime magic squares

PlanetMath, Prime magic squares

Stefano Tognon, Prime Analysis

Example

From Natalia Makarova, Sep 26 2009: (Start)
Here is a 14 X 14 example:
  [  3   43   59  131  181  271  383  599  797  919  971 1039 1123 1193
  1151  433  967  211  337  491  397  691   83  523  593  773  449  613
   263  373  101 1063  877  617  419  911  787  241  151  839  739  331
   503  439  809 1051 1091  659  157 1031   71  139  379  179  743  461
   173  647 1069  389 1049   19  311  223  317 1103  283  947  499  683
   547   13 1061  353  229  853  677  751  571  983 1201   29  193  251
   643  269  887  733   23  409 1129  191  769  401   47 1109  149  953
   163  881  673  107  431  487  991  631  829  109  349  367  811  883
  1163  827  607 1171  443  653  463    5  457  577   31  293  601  421
   509 1097  313  757  167  709  761  347  857  137  619  233   89 1117
  1093 1019    7  521 1033   61   73  941 1009  859  701   11  127  257
    53  467   97  307 1153  557 1021  569  359  937  821  113  977  281
   907   17  823  641  661  929   67  719   79  587  479  563 1013  227
   541 1187  239  277   37  997  863  103  727  197 1087 1217  199   41 ]
(End)
Comment from N. J. A. Sloane, Sep 28 2009: this contains 192 consecutive primes, 3 to 1171, plus 1187, 1193, 1201, 1217.
For the 3 X 3 case see A024351. For the 4 X 4 magic square see the Mathworld link.

Crossrefs

Cf. A073502, A073350, A125007.

Sequence in context: A159442 A046189 A278198 * A268790 A077786 A188537

Adjacent sequences: A164840 A164841 A164842 * A164844 A164845 A164846

Keyword

nonn,more

Author

Andrew Lelechenko, Aug 28 2009 and Natalia Makarova, Sep 08 2009

Extensions

Partially reworded by R. J. Mathar, Aug 31 2009

Edited by N. J. A. Sloane, Sep 14 2009

a(11)-a(15) from Natalia Makarova, a(16)-a(35) from Natalia Makarova and Stefano Tognon

Edited by Max Alekseyev, Feb 11 2010

Status

approved