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
Examples
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]
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
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.
Programs
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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
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
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