A189188 Potential magic constants of 8 X 8 magic squares composed of consecutive primes.

2016, 2244, 2336, 2570, 2762, 4106, 4362, 4464, 4566, 4670, 4776, 4934, 5952, 6870, 7036, 7146, 7588, 7644, 7700, 8824, 9756, 9930, 9988, 10394, 10454, 10514, 10690, 10868, 10928, 11560, 12620, 12682, 14986, 15424, 15808, 16000, 16510, 18668, 20434

Offset

1,1

Comments

For an 8 X 8 magic square composed of 64 consecutive primes, it is necessary that the sum of these primes is a multiple of 16.

This sequence consists of even integers equal to the sum of 64 consecutive primes divided by 8. It is not known whether each such set of consecutive primes can be arranged into an 8 X 8 magic square but it looks plausible.

From A.H.M. Smeets, Jan 20 2021: (Start)

Except from the condition that a magic constant exists, it must be an even magic constant due to the fact that the order is even, which explains why the sum of primes must be divisable by 16.

The number of possible combinations of 8 primes out of the 64 consecutive primes added results in the magic constant is such that in almost all cases such a magic square existsts. However, as n increases, the diversity in prime gaps between the 64 consecutive primes increases, and thus the probability that a potential magic constant will lead to a magic square configuration will decrease. The challenge here seems to be to find a potential magic constant which has no magic square configuration. (End)

Links

A.H.M. Smeets, Table of n, a(n) for n = 1..20000

Natalia Makarova Magic squares of order 8 which consist of consecutive primes (in Russian)

A.H.M. Smeets, Magic squares of consecutive primes of order 8

Example

a(1) = 2016
  [ 79 137 197 199 277 347 349 431
   127 193 131 419 337 421 107 281
   103 379 283 389 293 227 179 163
   397 251  83 271 269 157 439 149
   409 211 383 191 181 101 401 139
   307 239 317 167  89 367  97 433
   353 233 359 151 257 223 331 109
   241 373 263 229 313 173 113 311 ]
.
a(12) = 4934
  [ 823 619 461 457 631 587 599 757
    443 563 647 509 733 761 787 491
    503 809 419 701 661 797 487 557
    683 499 743 677 449 607 617 659
    439 727 571 577 719 821 601 479
    811 641 593 523 421 467 709 769
    691 433 673 751 773 431 613 569
    541 643 827 739 547 463 521 653 ]

Maple

s:= proc(n) option remember;
       `if` (n=1, add (ithprime(i), i=1..64),
                  ithprime(n+63) -ithprime(n-1) +s(n-1))
    end:
a:= proc(n) option remember; local k, m; a(n-1);
       for k from 1+b(n-1) while irem(s(k), 16, 'm')<>0 do od;
       b(n):= k; 2*m
    end:
a(0):=0: b(0):=0:
seq(a(n), n=1..50);

Crossrefs

Cf. A073520, A173981, A176571, A177434, A188536, A191679, A192087.

Sequence in context: A076666 A323470 A309918 * A076582 A323749 A323469

Adjacent sequences: A189185 A189186 A189187 * A189189 A189190 A189191

Keyword

nonn

Author

Natalia Makarova, Apr 18 2011

Extensions

Edited by Max Alekseyev, Jun 18 2011

Status

approved