A073523 -id:A073523 - cp's OEIS frontend

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

N. J. A. Sloane, Aug 29 2002

			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]

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.

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

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.

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

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

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

A073519 The set of nine consecutive primes forming a 3 X 3 magic square with the smallest magic constant (4440084513).

Original entry on oeis.org

1480028129, 1480028141, 1480028153, 1480028159, 1480028171, 1480028183, 1480028189, 1480028201, 1480028213

Offset: 1

N. J. A. Sloane, Aug 29 2002

The square is given (with the terms in correct order) in A320873. The (increasingly ordered) set of primes does not contain more information than the magic constant (= sum) S, since they have to be consecutive and sum up to 3*S. It is easy to construct the unique set of (consecutive) primes with this property, cf. PROGRAM. - M. F. Hasler, Oct 28 2018

			The magic square is
[ 1480028201 1480028129 1480028183 ]
[ 1480028153 1480028171 1480028189 ]
[ 1480028159 1480028213 1480028141 ]

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.

Harvey D. Heinz, Prime Numbers Magic Squares: Minimum consecutive primes - 3, 1999-2010.
Index entries for sequences related to magic squares

Cf. A024351, A073520, A073521, A073522, A073523, A256891, A265139, A265614.

A073519=MagicPrimes(4440084513,3) \\ where: (also used in A073521, ...)
MagicPrimes(S, n, P=[nextprime(S\n)])={S=n*S-P[1]; for(i=1, -1+n*=n, S-=if(S>(n-i)*P[1], P=concat(P, nextprime(P[#P]+1)); P[#P], P=concat(precprime(P[1]-1), P); P[1])); if(S, -P, P)} \\ The vector of n^2 primes whose sum is n*S, or a negative vector with an approximate solution if no exact solution exists. - M. F. Hasler, Oct 22 2018

A073521 The set of 16 consecutive primes with the property that they form a 4 X 4 magic square with the smallest magic constant (258).

Original entry on oeis.org

31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101

Offset: 1

N. J. A. Sloane, Aug 29 2002

			The magic square is
[ 37 83 97 41 ]
[ 53 61 71 73 ]
[ 89 67 59 43 ]
[ 79 47 31 101 ]

Allan W. Johnson, Jr., Journal of Recreational Mathematics, vol. 14:2, 1981-82, pp. 152-153.
Clifford A. Pickover, The Zen of Magic Squares, Circles and Stars: An Exhibition of Surprising Structures across Dimensions, Princeton University Press, 2002.

Harvey Heinz, Prime Magic Squares
Bill McEachen, Python program
Index entries for sequences related to magic squares

Cf. A073519, A073520, A073522, A073523.

A073521=MagicPrimes(258,4) \\ See A073519 for MagicPrimes(). - M. F. Hasler, Oct 28 2018

A073522 A set of 25 consecutive primes that form a 5 X 5 magic square with the (non-minimal) magic constant 1703.

Original entry on oeis.org

269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419

Offset: 1

N. J. A. Sloane, Aug 29 2002

The magic constant here is not the smallest possible for a 5 X 5 magic square composed of consecutive primes, this would be A073520(5) = 313 corresponding to primes (13, 17, ..., 113). [Edited by M. F. Hasler, Oct 29 2018]

			The magic square is
[ 281 409 311 419 283 ]
[ 359 379 349 347 269 ]
[ 313 307 389 293 401 ]
[ 397 331 337 271 367 ]
[ 353 277 317 373 383 ]

Allan W. Johnson, Jr., Journal of Recreational Mathematics, vol. 14:2, 1981-82, pp. 152-153.
Clifford A. Pickover, The Zen of Magic Squares, Circles and Stars: An Exhibition of Surprising Structures across Dimensions, Princeton University Press, 2002.

Harvey Heinz, Prime Magic Squares
Index entries for sequences related to magic squares

Cf. A073519 and A320873 (minimal 3 X 3 magic square of consecutive primes), A073520 (minimal magic sum for n X n square of consecutive primes), A073521 (consecutive primes of a 4 X 4 magic square), A073523 (consecutive primes of a pandiagonal 6 X 6 magic square).

A073522=MagicPrimes(1703,5) \\ Cf. A073519. - M. F. Hasler, Oct 28 2018

Edited by Max Alekseyev, Sep 24 2009

A104157 Smallest of n^2 consecutive primes that form an n X n magic square with the least magic constant, or 0 if no such magic square exists.

Original entry on oeis.org

2, 0, 1480028129, 31, 13, 7, 7, 79, 37, 23, 67, 89, 13, 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

Offset: 1

Robert G. Wilson v, Mar 09 2005

The magic constants (= sums) are given in A073520. For a given sum, the corresponding list of primes (and thus also the smallest one) is easily calculated, cf. PARI code. - M. F. Hasler, Oct 29 2018

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.

Harvey Heinz, Prime Magic Squares
Stefano Tognon, Table for prime magic squares
Index entries for sequences related to magic squares

Cf. A073519 or A320873 (the square for 3 X 3), A073520 (magic sums for 4 X 4 squares of consecutive primes), A073521 (consecutive primes of a 4 X 4 magic square), A073522 (consecutive primes of a (non minimal!) 5 X 5 magic square), A073523 (consecutive primes of a pandiagonal 6 X 6 magic square).

A104157(n)=MagicPrimes(A073520[n],n)[1] \\ See A073519 for MagicPrimes(). This code uses a precomputed array A073520, but in practice one would rather compute that sequence as function of this one. - M. F. Hasler, Oct 29 2018

Conjecture: for n > 4, a(n) = prime(s) where s > 1 is the smallest integer such that (Sum_{i=s..s+n^2-1} prime(i))/n is an integer of the same parity as n. - Max Alekseyev, Jan 29 2010

a(n) = prime(i) such that Sum_{k=0..n^2-1} prime(i+k) = n*A073520(n). - M. F. Hasler, Oct 29 2018

a(5)-a(6) corrected, a(7)-a(20) added by Max Alekseyev, Sep 24 2009
Definition edited by N. J. A. Sloane, Oct 03 2009
More terms from Max Alekseyev, Jan 29 2010

A177434 The magic constants of 6 X 6 magic squares composed of consecutive primes.

Original entry on oeis.org

484, 744, 806, 868, 930, 1390, 1460, 1494, 1634, 1704, 1740, 1848, 1992, 2100, 2172, 2316, 2390, 2540, 3116, 3192, 3694, 3734, 3774, 4486, 4946, 4988, 5736, 6104, 6148, 6526, 6568, 6610, 6776, 6820, 6950, 7036, 7078, 7120, 7984, 8118, 8162, 8828, 9318

Offset: 1

Natalia Makarova, May 08 2010

nonn

Let Z be a sum of 36 consecutive primes. A necessary condition to get a 6 X 6 magic square using these primes is that Z=6S, where S is even. The smallest magic constant of a 6 X 6 magic square of consecutive primes is 484 (cf. A073520).
Each of the first 100 possible arrays of 36 consecutive primes which satisfy the necessary condition produces a magic square.
A program written by Stefano Tognon was used.

			S = 744
   [139 113 151 131  83 127]
   [223 149  89  47 157  79]
   [173 103 181 167  59  61]
   [ 67 137  53  97 211 179]
   [101 199  73 109  71 191]
   [ 41  43 197 193 163 107]
S = 806
   [131  53 107 157 191 167]
   [ 89 229 179  97 109 103]
   [ 83 211  71 139  79 223]
   [113 101 137 181 227  47]
   [197  61 163  59 127 199]
   [193 151 149 173  73  67]
S = 868
   [191 137  79 193 197  71]
   [ 67 157  73 229 239 103]
   [179 173 167  97 101 151]
   [211 181 223  61 109  83]
   [113 131 199 139  59 227]
   [107  89 127 149 163 233]
Magic square with S=930 can be pan-diagonal (cf. A073523).
Example of a non-pan-diagonal square:
S = 930
   [167  71 151 199 131 211]
   [ 89 241 181  73 113 233]
   [ 83 227 127 197 229  67]
   [239 137 139 103 163 149]
   [179  97 223 251 101  79]
   [173 157 109 107 193 191]

Natalya Makarova, Author's webpage (in Russian)

Cf. A173981 (analog for 4 X 4), A176571 (analog for 5 X 5), A073523 (36 consecutive primes of a pandiagonal magic square), A073520 (smallest magic sum for n X n), A259733 (most-perfect 8 X 8), A272387 (smallest element of 6 X 6 magic squares of consecutive primes).

A177434(n, p=A272387[n], N=6)=sum(i=2, N^2, p=nextprime(p+1), p)/N \\ Uses a precomputed array A272387, but can actually be used to find the terms, cf A272387. - M. F. Hasler, Oct 28 2018

a(n) = Sum_{k=0..35} A000040(A000720(A272387(n))+k)/6. - M. F. Hasler, Oct 28 2018

Edited by M. F. Hasler, Oct 28 2018

A245721 The set of 16 consecutive primes forming a 4 X 4 pandiagonal magic square with the smallest magic constant, 682775764735680 = A256234(1).

Original entry on oeis.org

170693941183817, 170693941183847, 170693941183859, 170693941183861, 170693941183889, 170693941183891, 170693941183903, 170693941183907, 170693941183933, 170693941183937, 170693941183949, 170693941183951, 170693941183979, 170693941183981, 170693941183993, 170693941184023

Offset: 1

Max Alekseyev, Jul 30 2014

Also, the set of 16 smallest consecutive primes forming a 4x4 Stanley antimagic square.

The set of primes is uniquely and straightforwardly determined by the magic sum, here A256234(1), cf. PROGRAM. See A320874 for the ordered list, i.e., the lexicographic smallest magic square made of these primes. - M. F. Hasler, Oct 23 2018

			A pandiagonal magic square formed by these primes:
  170693941183817 170693941183933 170693941183949 170693941183981
  170693941183979 170693941183951 170693941183847 170693941183903
  170693941183891 170693941183859 170693941184023 170693941183907
  170693941183993 170693941183937 170693941183861 170693941183889
A Stanley antimagic square formed by these primes:
  170693941183817 170693941183859 170693941183907 170693941183949
  170693941183847 170693941183889 170693941183937 170693941183979
  170693941183861 170693941183903 170693941183951 170693941183993
  170693941183891 170693941183933 170693941183981 170693941184023

Index entries for sequences related to magic squares

Cf. A320874 (the square made of the set of primes given here).
Cf. A073519 or A320873, A073521, A073522 (3 X 3, 4 X 4 and 5 X 5 consecutive primes), A073523 and A320876 (6 X 6 consecutive primes, pandiagonal magic square).
Cf. A210710: Minimal index of a Stanley antimagic square of order n consisting of distinct primes.
Cf. A073520: Smallest magic sum of a magic square made of n^2 consecutive primes.
Cf. A104157: Smallest of n X n consecutive primes forming a magic square.
Cf. A256234: Magic sums of 4 X 4 pandiagonal magic squares of consecutive primes.

A245721=MagicPrimes(682775764735680,4) \\ See A073519.

A320876 Lexicographically first 6 X 6 pandiagonal magic square made of consecutive primes with the smallest magic constant (930).

Original entry on oeis.org

67, 139, 241, 73, 199, 211, 193, 233, 97, 167, 103, 137, 71, 113, 191, 131, 227, 197, 251, 181, 89, 229, 101, 79, 109, 157, 163, 151, 127, 223, 239, 107, 149, 179, 173, 83

Offset: 1

M. F. Hasler, Oct 22 2018

The same 6 X 6 terms are given in increasing order in sequence A073523. But giving them in increasing order does not contain more information as the smallest of them or magic constant (= sum) itself, which uniquely determines the sequence of primes since they have to be consecutive and their sum is equal to 6 times the magic constant. The present sequence gives the full information about the magic square.
A pandiagonal magic square allows rotations (rather than arbitrary cyclic permutations) of columns or rows, as well as reflection on the 4 symmetry axes of the square. Considering all these variants of this square, there is none with elements coming earlier than (67, 139, ...)
There exist non-pandiagonal 6 X 6 magic squares composed of consecutive primes with smaller magic constant, the smallest being A073520(6) = 484.
Pandiagonal means that not only the 2 main diagonals, but all other 10 diagonals also have the same sum, Sum_{i=1..6} A[i,M6(k +/- i)] = 930 for k = 1, ..., 6 and M6(x) = y in {1, ..., 6} such that y == x (mod 6).

			The magic square is
  [ 67 139 241  73 199 211]
  [193 233  97 167 103 137]
  [ 71 113 191 131 227 197]
  [251 181  89 229 101  79]
  [109 157 163 151 127 223]
  [239 107 149 179 173  83]

Allan W. Johnson, Jr., Journal of Recreational Mathematics, vol. 23:3, 1991, pp. 190-191.
Clifford A. Pickover, The Zen of Magic Squares, Circles and Stars: An Exhibition of Surprising Structures across Dimensions, Princeton University Press, 2002.

Harvey Heinz, Prime Magic Squares
Index entries for sequences related to magic squares

Cf. A073519, A073520, A073521, A073522.

/* the following transformation operators for matrices, together with transposition, allow the production of all variants of a (pandiagonal) magic square */
REV(M)=matconcat(Vecrev(M)) \\ reverse the order of columns of M
FLIP(M)=matconcat(Colrev(M)) \\ reverse the order of row of M
ROT(M,k=1)=matconcat([M[,k+1..#M],M[,1..k]]) \\ rotate left by k (default: 1) columns
ALL(M)=Set(concat(apply(M->vector(#M,k,ROT(M,k)),[M,M~,REV(M),REV(M~),FLIP(M),FLIP(M~)]))) \\ PARI orders the set according to the (first) columns of the matrices, so one must take the transpose to get them ordered according to elements of the first row.
MagicPrimes(S=930,n=6,P=[nextprime(S\n)])={S=n*S-P[1];for(i=1,-1+n*=n,S-=if(S>(n-i)*P[1],P=concat(P,nextprime(P[#P]+1));P[#P],P=concat(precprime(P[1]-1),P);P[1]));if(S,-P,P)} \\ The vector of n^2 primes whose sum is n*S (= A073523 for default values), or a negative vector of "best approximation" if there is no exact solution.

A320874 Lexicographically first 4 X 4 pandiagonal magic square made of consecutive primes.

Original entry on oeis.org

170693941183817, 170693941183933, 170693941183949, 170693941183981, 170693941183979, 170693941183951, 170693941183847, 170693941183903, 170693941183891, 170693941183859, 170693941184023, 170693941183907, 170693941183993, 170693941183937, 170693941183861, 170693941183889

Offset: 1

M. F. Hasler, Oct 22 2018

This is also the 4 X 4 pandiagonal magic square made of consecutive primes which has the smallest possible magic constant (= sum), 682775764735680 = A256234(1). (In the present case there is no other non-equivalent pandiagonal 4 X 4 magic square having the same magic sum, but this could be possible as for rows 7 and 8 of A320872.)
There exist many non-pandiagonal 4 X 4 magic squares composed of consecutive primes with much smaller magic constant, the smallest being A073520(4) = 258.
Pandiagonal means that not only the 2 main diagonals, but also the 6 other "broken" diagonals all have the same sum, Sum_{i=1..4} A[i,M4(k +- i)] = 682775764735680 for k = 1, ..., 4 and M4(x) = y in {1, ..., 4} such that y == x (mod 4).
A pandiagonal magic square allows rotations (but not arbitrary cyclic permutations like, e.g., 1 -> 3 -> 4 -> 1) of columns or rows, as well as reflection on the 4 symmetry axes of the square (which also produce rotations of 90 degrees around the center of the square). Among all these variants of this square, there is none with elements coming earlier than (170693941183817, 170693941183933, ...), cf. PROGRAM for explicit verification.
The same 4 X 4 primes are given in increasing order in sequence A245721. But does not give more information than smallest term, the central term, or the magic constant itself (cf. A256234) which uniquely determines the sequence of primes (cf. PARI code) since they have to be consecutive and their sum is equal to 4 times the magic constant. The present sequence gives the full information about the magic square, and the given PARI code allows the production of all "equivalent" variants of the square.

			The magic square is
  [ 170693941183817 170693941183933 170693941183949 170693941183981 ]
  [ 170693941183979 170693941183951 170693941183847 170693941183903 ]
  [ 170693941183891 170693941183859 170693941184023 170693941183907 ]
  [ 170693941183993 170693941183937 170693941183861 170693941183889 ]

Allan W. Johnson, Jr., Journal of Recreational Mathematics, vol. 23:3, 1991, pp. 190-191.
Clifford A. Pickover, The Zen of Magic Squares, Circles and Stars: An Exhibition of Surprising Structures across Dimensions, Princeton University Press, 2002.

Harvey Heinz, Prime Magic Squares
Index entries for sequences related to magic squares

Cf. A073519 and A320873, A073521, A073522 (3 X 3, 4 X 4 and 5 X 5 consecutive primes), A073523 and A320876 (6 X 6 consecutive primes, pandiagonal magic square).
Cf. A210710: Minimal index of a Stanley antimagic square of order n consisting of distinct primes.
Cf. A073520: Smallest magic sum for an n^2 magic square made of consecutive primes.
Cf. A104157: Smallest of n X n consecutive primes forming a magic square.
Cf. A256234: Magic sums of 4 X 4 pandiagonal magic squares of consecutive primes.

/* the following transformation operators for matrices, together with transposition, allow the production of all (24 for n=4) variants of a (pandiagonal) magic square */
REV(M)=matconcat(Vecrev(M)) \\ reverse the order of columns of M
FLIP(M)=matconcat(Colrev(M)) \\ reverse the order of rows of M
ROT(M,k=1)=matconcat([M[,k+1..#M],M[,1..k]]) \\ rotate left by k (default: 1) columns
ALL(M)=Set(concat(apply(M->vector(#M,k,ROT(M,k)),[M,M~,REV(M),REV(M~),FLIP(M),FLIP(M~)]))) \\ PARI orders the set according to the (first) columns of the matrices, so one must take the transpose to get them ordered according to elements of the first row.
\\ The set of primes is A245721=MagicPrimes(682775764735680,4), cf. A073519.

A179440 The smallest magic constant of pan-diagonal magic squares which consist of distinct prime numbers.

Original entry on oeis.org

240, 395, 450, 733

Offset: 4

Natalia Makarova, Jul 14 2010

Classic pan-diagonal magic squares exist for orders n > 3 not of the form 4k+2.
Non-traditional pandiagonal magic squares exist for all orders n > 3.
Bounds for further terms: a(8) <= 1248, a(9) <= 2025, a(10) <= 2850, a(11) <= 4195, a(12) <= 5544, a(13) <= 7597.

			a(5) = 395 (found by V. Pavlovsky)
    5  73 127 137  53
   37 167  17  71 103
   83 101  13  67 131
   43  31 197 113  11
  227  23  41   7  97
.
a(6) = 450 (found by Radko Nachev)
    3   5  89 137  67 149
  127 163   7  29  11 113
   31  23 167  59 157  13
  107  97  43  53 131  19
   73  79  41  71  47 139
  109  83 103 101  37  17
.
a(7) = 733 (found by Jarek Wroblewski)
    3   7 173 223  17 197 113
  181 211  11  79 131  23  97
   43  41 149  89 137 191  83
  233 103 107  73 127  31  59
   29 167 101  19 199  67 151
    5  47 139 179 109  61 193
  239 157  53  71  13 163  37

N. Makarova, (in Russian)
V. Pavlovsky, (in Russian)
S. Belyaev , (in Russian)
Mutsumi Suzuki, MagicSquare
Al Zimmermann's Programming Contests, Pandiagonal Magic Squares of Prime Numbers: Final Report

Cf. A073523

Correction for the third term with example given Natalia Makarova, Jul 21 2010
Link and example corrected by Natalia Makarova, Aug 01 2010
Edited by Max Alekseyev, Mar 15 2011
Bound for a(9) improved by Alex Chernov, Apr 23 2011
Bound for a(12) improved by Natalia Makarova, Jun 21 2011
Corrected a(6) from Radko Nachev, added by Max Alekseyev, May 28 2013
a(7) from Jarek Wroblewski and new bounds from Al Zimmermann's contest, added by Max Alekseyev, Oct 11 2013

cp's OEIS Frontend

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

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A073519 The set of nine consecutive primes forming a 3 X 3 magic square with the smallest magic constant (4440084513).

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A073521 The set of 16 consecutive primes with the property that they form a 4 X 4 magic square with the smallest magic constant (258).

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A073522 A set of 25 consecutive primes that form a 5 X 5 magic square with the (non-minimal) magic constant 1703.

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A104157 Smallest of n^2 consecutive primes that form an n X n magic square with the least magic constant, or 0 if no such magic square exists.

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A177434 The magic constants of 6 X 6 magic squares composed of consecutive primes.

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A245721 The set of 16 consecutive primes forming a 4 X 4 pandiagonal magic square with the smallest magic constant, 682775764735680 = A256234(1).

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