A073520 -id:A073520 - cp's OEIS frontend

A192087 Potential magic constants of a 10 X 10 magic square composed of consecutive primes.

2862, 3092, 3500, 4222, 4780, 5608, 7124, 10126, 10198, 11212, 11426, 12140, 12212, 12284, 12356, 12428, 12714, 12854, 12924, 15270, 16252, 16476, 18594, 18672, 18750, 18828, 19214, 20764, 21150, 23752, 24214, 24598, 24828, 27180, 27342, 27424, 27916, 28666, 29406, 29568

Offset: 1

Natalia Makarova, Jun 23 2011

nonn

For a 10 X 10 magic square composed of 100 consecutive primes, the sum of these primes must be a multiple of 20.
This sequence consists of even integers equal the sum of 100 consecutive primes divided by 10. It is not known whether each such set of consecutive primes can be arranged into a 10 X 10 magic square but it looks plausible.
Actual magic squares were constructed for all listed magic constants <= 11212.

			a(1)=2862 for a square containing prime(9)..prime(108):
  [23  179  409  373  263  137  461  457  523   37
  193  353  443  199  317  109  337  397  131  383
   71   73  389  251  593  167  439  449  233  197
  571  293  101  229   29  557  271   31  379  401
  127  419  283  241  269  239  547   89  181  467
  491  433  223  113   41  577   43  311  563   67
  281   97  163  587  191  313  149  509  421  151
  307  499  227  431  103   83   59  479  211  463
  277  359  257  331  569  541   53   79   47  349
  521  157  367  107  487  139  503   61  173  347]
.
a(10)=11212
  [769   863  1171   967   859  1381  1237  1459  1289  1217
  1163   953   797  1297  1049  1021  1303   977  1423  1229
   809  1277  1153   937  1151  1409  1291   839  1249  1097
  1429  1231  1193  1451  1061   829   821  1361   823  1013
  1453   997   947  1091  1321   887  1283   941   811  1481
  1069  1201  1427  1129   907   919  1373  1039  1117  1031
  1009  1123  1301  1093  1367  1483   911  1051  1087   787
   991  1109  1279   877  1223   929  1187  1433  1327   857
  1213  1439  1063   971  1447   883   773  1259   983  1181
  1307  1019   881  1399   827  1471  1033   853  1103  1319]

Natalia Makarova, Sequence of Magic Numbers MK 10th Order (in Russian).

Cf. A073520, A173981, A176571, A177434, A188536, A189188, A191679.

s:= proc(n) option remember;
       `if` (n=1, add (ithprime(i), i=1..100),
                  ithprime(n+99) -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), 20, 'm')<>0 do od;
       b(n):= k; m
    end:
a(0):=0: b(0):=0:
seq (2*a(n), n=1..50);

A270829 Smallest magic sum for any n X n semimagic square made from consecutive primes, or 0 if no such magic square exists.

Original entry on oeis.org

2, 0, 65573, 124, 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

Offset: 1

Arkadiusz Wesolowski, Mar 23 2016

nonn

a(n) <= A073520(n). For n = 3 and 4, a(n) is different from A073520(n).

Index entries for sequences related to magic squares

Cf. A073520, A268912, A270830.

a(2) = 0, otherwise a(n) = (1/n) * sum(m=k..n^2+k-1, A000040(m)), where k = A049084(A270830(n)).

A342124 a(n) = smallest potential magic constant for an n X n magic square composed of consecutive primes.

Original entry on oeis.org

2, 18, 121, 124, 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

Offset: 1

A.H.M. Smeets, Feb 28 2021

nonn

A potential magic constant is defined by the necessary condition that for any potential magic square the sum of the consecutive primes must be divisible by the order n, and, after dividing the sum of consecutive primes by the order n, it must have the same parity as the order n.

The known smallest magic constants for any n X n magic square composed of consecutive primes are given by A073520. It is believed that for n >= 5, a(n) = A073520(n).

Cf. A073520, A104157, A342386.

A342386 Smallest prime out of n^2 consecutive primes such that this set of consecutive primes has the smallest potential magic constant (as given in A342124).

Original entry on oeis.org

2, 5, 23, 5, 13, 7, 7, 79, 37, 23, 67, 89, 131, 31, 71, 47, 43, 73, 277, 353, 41, 67, 127, 223, 79, 13, 193, 5, 23, 43, 5, 67, 3, 19, 5, 59, 59, 653, 19, 19, 97, 409, 5, 383, 29, 137, 379, 349, 653, 1187, 47, 41, 37, 17, 619, 89, 283, 283, 43, 479, 191, 1009, 571

Offset: 1

A.H.M. Smeets, Mar 10 2021

nonn

A potential magic constant is defined by the necessary condition to obtain a magic square, that for any potential magic square the sum of the consecutive primes must be divisible by the order n, and, after dividing the sum of consecutive primes by the order n, it must have the same parity as the order n.

The known smallest primes out of a set of n^2 consecutive primes such that its magic constant is the smallest possible magic constant as well, are given by A104157. It is believed that for n >= 5, a(n) = A104157(n).

Cf. A073520, A104157, A342124.

A368676 Number of 4 X 4 prime magic squares with magic sum 2n.

Original entry on oeis.org

128, 0, 0, 160, 0, 0, 224, 0, 64, 384, 64, 192, 2112, 96, 224, 4768, 0, 480, 5472, 160, 1088, 6688, 160, 1632, 13600, 416, 1728, 24640, 544, 3008, 40736, 512, 6720, 45504, 672, 11776, 41984, 2752, 17888, 65760, 4416, 18688, 128544, 4544, 21888, 162240, 3712

Offset: 60

Zhao Hui Du, Jan 02 2024

nonn

			[17, 11, 31, 61]
[43, 67,  3,  7]
[41, 37, 13, 29]
[19,  5, 73, 23]
 is a 4 X 4 prime magic square in which the elements in each row and column and both diagonals sum to 120 and all elements are prime numbers. There are a total of 128 such prime magic squares so a(60)=128.

Eric Weisstein's World of Mathematics, Prime Magic Square.

Cf. A006052, A073520, A073521, A271578.

cp's OEIS Frontend

A192087 Potential magic constants of a 10 X 10 magic square composed of consecutive primes.

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A270829 Smallest magic sum for any n X n semimagic square made from consecutive primes, or 0 if no such magic square exists.

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A342124 a(n) = smallest potential magic constant for an n X n magic square composed of consecutive primes.

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A342386 Smallest prime out of n^2 consecutive primes such that this set of consecutive primes has the smallest potential magic constant (as given in A342124).

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A368676 Number of 4 X 4 prime magic squares with magic sum 2n.

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