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%I #14 Jan 09 2021 02:12:28
%S 177,120,233,432,733,1154,1731,2470,3417,4584,6013,7712,9731,12088,
%T 14807,17940,21501,25530,30021,35086,40675,46840,53631,61092,69251,
%U 78100,87697,98084,109309,121380,134377,148258,163043
%N The smallest magic constant of an n X n magic square with distinct prime entries.
%C 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.
%H Harvey Heinz, <a href="http://www.magic-squares.net/primesqr.htm">Prime magic squares</a>
%H A. Lelechenko and N. Makarova, <a href="/A164843/a164843.txt">Examples of prime magic n X n squares with minimal magic constant for n=5..13.</a>
%H N. Makarova, <a href="http://www.natalimak1.narod.ru/sqmin1.htm">Smallest prime magic squares, Part I</a> (in Russian)
%H N. Makarova, <a href="http://www.natalimak1.narod.ru/sqmin2.htm">Smallest prime magic squares, Part II</a> (in Russian)
%H Mathworld, <a href="http://mathworld.wolfram.com/PrimeMagicSquare.html">Prime magic squares</a>
%H PlanetMath, <a href="http://planetmath.org/encyclopedia/PrimeMagicSquare.html">Prime magic squares</a>
%H Stefano Tognon, <a href="http://digilander.libero.it/ice00/magic/prime/PrimeAnalysis.html">Prime Analysis</a>
%e From _Natalia Makarova_, Sep 26 2009: (Start)
%e Here is a 14 X 14 example:
%e [ 3 43 59 131 181 271 383 599 797 919 971 1039 1123 1193
%e 1151 433 967 211 337 491 397 691 83 523 593 773 449 613
%e 263 373 101 1063 877 617 419 911 787 241 151 839 739 331
%e 503 439 809 1051 1091 659 157 1031 71 139 379 179 743 461
%e 173 647 1069 389 1049 19 311 223 317 1103 283 947 499 683
%e 547 13 1061 353 229 853 677 751 571 983 1201 29 193 251
%e 643 269 887 733 23 409 1129 191 769 401 47 1109 149 953
%e 163 881 673 107 431 487 991 631 829 109 349 367 811 883
%e 1163 827 607 1171 443 653 463 5 457 577 31 293 601 421
%e 509 1097 313 757 167 709 761 347 857 137 619 233 89 1117
%e 1093 1019 7 521 1033 61 73 941 1009 859 701 11 127 257
%e 53 467 97 307 1153 557 1021 569 359 937 821 113 977 281
%e 907 17 823 641 661 929 67 719 79 587 479 563 1013 227
%e 541 1187 239 277 37 997 863 103 727 197 1087 1217 199 41 ]
%e (End)
%e Comment from _N. J. A. Sloane_, Sep 28 2009: this contains 192 consecutive primes, 3 to 1171, plus 1187, 1193, 1201, 1217.
%e For the 3 X 3 case see A024351. For the 4 X 4 magic square see the Mathworld link.
%Y Cf. A073502, A073350, A125007.
%K nonn,more
%O 3,1
%A _Andrew Lelechenko_, Aug 28 2009 and _Natalia Makarova_, Sep 08 2009
%E Partially reworded by _R. J. Mathar_, Aug 31 2009
%E Edited by _N. J. A. Sloane_, Sep 14 2009
%E a(11)-a(15) from _Natalia Makarova_, a(16)-a(35) from _Natalia Makarova_ and Stefano Tognon
%E Edited by _Max Alekseyev_, Feb 11 2010
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