cp's OEIS Frontend

A necessary condition for a prime being in this sequence is that the sum of this and the subsequent 24 primes divided by 5 must be an odd integer. - M. F. Hasler, Oct 30 2018

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

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).

See A320871, A320872 and A320873 for the list of all 3 X 3 magic squares of distinct integers, primes, resp. consecutive primes. In all these, only the lexicographically smallest of the eight "equivalent" squares are listed. Note that the terms are not always in the order that corresponds to the terms of this sequence. For example, in row 3 of A320871 and row 11 of A320873, the second term is smaller than the third term. However, when this is not the case, then row n of the present sequence is the list of indices which gives the n-th variant of the square from the (ordered) set of 9 elements: e.g., (2, 7, 6, ...) means that the 2nd, 7th and 6th of the set of 9 numbers yield the first row of the square. For example, A320873(n) = A073519(a(n)), 1 <= n <= 9. - M. F. Hasler, Nov 04 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.

1st row: -53 157 7

2nd row: 97 37 -23

3rd row: 67 -83 127

Magic 3 x 3 square with minimal magic constant M = 3 * 37 = 111.

Elements are in arithmetic 1st-order progression, negative p are primes.

This magic prime square (MPS) from the author dates from 2008.

Note that Dudeney's MPS from 1917 has the nonprime element 1:

67 1 43

13 37 61

31 73 7

Madachy/Ondrejka gave 1979 MPS with M = 3 * 59 = 177 > 111:

17 89 71

113 59 5

47 29 101