A073520 Smallest magic constant for any n X n magic square made from consecutive primes, or 0 if no such magic square exists.

2, 0, 4440084513, 258, 313, 484, 797, 2016, 2211, 2862, 4507, 6188, 6325, 9660, 12669, 13016, 16857, 19530, 23069, 28184, 38761, 46302, 42515, 49846, 59087, 70260, 73385, 78960, 97267, 98316, 111023, 124454, 134641, 152952, 163043, 180596, 195975, 218432, 237623, 293182, 276243, 298868

Offset

1,1

References

Allan W. Johnson, Jr., Journal of Recreational Mathematics, vol. 14:2, 1981-82, pp. 152-153.

Allan W. Johnson, Jr., Journal of Recreational Mathematics, vol. 23:3, 1991, pp. 190-191.

H. L. Nelson, Journal of Recreational Mathematics, 1988, vol. 20:3, p. 214.

Clifford A. Pickover, The Zen of Magic Squares, Circles and Stars: An Exhibition of Surprising Structures across Dimensions, Princeton University Press, 2002.

Links

M. F. Hasler, Table of n, a(n) for n = 1..63

Mutsumi Suzuki, Study of Magic Squares, 1957, in Japanese. Gives minimal squares of orders from 4 to 9 composed of consecutive primes.

Harvey Heinz, Prime Magic Squares

N. Makarova Squares 7x7, 8x8, 9x9, Squares 10x10, 11x11, 12x12, Square 14x14 (in Russian)

Stefano Tognon, Table for prime magic squares

Eric Weisstein's World of Mathematics, Prime Magic Square

Index entries for sequences related to magic squares

Formula

Conjecture: for n >= 5, a(n) equals the smallest integer of the form (A000040(s+1) + ... + A000040(s+n^2))/n = (A007504(s+n^2) - A007504(s))/n of the same parity as n.

a(2) = 0, otherwise a(n) = (1/n) * Sum_{m=k..n^2+k-1} A000040(m), where k = A049084(A104157(n). - Arkadiusz Wesolowski, Nov 06 2015

In the above, A049084 could be replaced by A000720 = primepi. - M. F. Hasler, Oct 29 2018

Example

A square of order 15 found by Natalia Makarova, communicated by Stefano Tognon, Sep 23 2009:
[  131  167  229  461  541  617  733  911  967 1091 1259 1279 1319 1471 1493
   547  907 1583 1613  149 1423  193 1601  941  137  233  389 1039 1283  631
  1019  181  751  163 1453 1301 1297 1277  271 1619 1327  691  277  281  761
  1307  719  359  919 1063  653 1237  269 1433  863 1439  313  191 1021  883
   503 1367  433 1013  829 1153  317  347 1109  491 1249  677 1451 1489  241
   421  311 1487  439 1049 1409 1123  463  409  983  449 1031 1163  373 1559
  1399 1193  419 1531  971  647  977 1051  709  479 1229  379  353 1093  239
   599  953 1213  587  499  727 1321  787  307 1151  157 1571 1033  773  991
   211 1291 1499  577 1087  349  947  467  739  613 1171 1609  173  839 1097
   563  139 1373 1459 1289  443  619 1201 1427  809  881 1303  331  263  569
   607 1607 1511  673 1181 1481 1217  523  661  857  223  743  197  431  757
   853  643  701  179 1483  571  769  859 1447  659  929  997 1223 1129  227
  1549  887  257  557  367 1061  601  337 1361  937 1231  811 1543  293  877
  1579 1187  397 1069  509  683  797 1567  401  383  641  283  823  827 1523
  1381 1117  457 1429  199  151  521 1009  487 1597  251  593 1553 1103 821]

Prog

(PARI) A073520(n, p=A104157[n])=sum(i=2, n^2, p=nextprime(p+1), p)/n \\ Assumes a pre-computed array A104157, but can be used to find a(n) and A104157(n) by calculating this for supplied primes p until the result satisfies the condition of the conjecture in FORMULA. - M. F. Hasler, Oct 29 2018

Crossrefs

Cf. A104157: smallest element in an n X n magic squares of consecutive primes.

Cf. A073519 and A320873 (3 X 3 magic square of consecutive primes), A073521 (consecutive primes of a 4 X 4 magic square), A245721 and A320874 (4 X 4 pandigital magic square of consecutive primes), A073522 (consecutive primes of a 5 X 5 magic square, non-minimal and non-pandiagonal), A073523 and A320876 (6 X 6 pandigital magic square of consecutive primes).

Cf. A256234: magic sums of 4 X 4 pandiagonal magic squares of consecutive primes.

Sequence in context: A270830 A270829 A104157 * A152137 A097470 A230093

Adjacent sequences: A073517 A073518 A073519 * A073521 A073522 A073523

Keyword

nonn,nice

Author

N. J. A. Sloane, Aug 29 2002

Extensions

a(5)-a(6) corrected and a(7)-a(14) added, from the work of Stefano Tognon and Natalia Makarova, by Max Alekseyev, Sep 23 2009

a(15) from Natalia Makarova, a(16) and further terms from Stefano Tognon

Edited by Max Alekseyev, Oct 13 2009

Edited and more terms (using A104157) from M. F. Hasler, Oct 29 2018

Status

approved