OFFSET
1,1
COMMENTS
The first row is the lexicographically first 3 X 3 magic square of consecutive primes with the smallest possible magic constant 4440084513 = A270305(1) = A073520(3).
The same 9 terms are also given in increasing order in sequence A073519. But this is equivalent of giving just the smallest of the terms (cf. A256891) or the central element (cf. A166113) or the magic constant itself (cf. A270305), which uniquely determines the sequence of primes since they have to be consecutive and their sum is equal to 3 times the magic constant.
In the case of 3 X 3 magic squares, however, the lexicographically smallest representative has its elements in a well-defined order, see comment in A320872. This allows the reconstruction of the square from the set of primes which can be computed from the central elements A166113 or magic constants A270305, cf. PROGRAM in A073519.
REFERENCES
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.
LINKS
EXAMPLE
The first row of 9 terms, (1480028141, 1480028189, 1480028183, 1480028213, 1480028171, 1480028129, 1480028159, 1480028153, 1480028201), corresponds to the following smallest 3 X 3 magic square of consecutive primes:
[1480028141 1480028189 1480028183]
[1480028213 1480028171 1480028129] .
[1480028159 1480028153 1480028201]
The eleventh row yields the first example where the second term is smaller than the third one:
[23813359643 23813359721 23813359727]
[23813359781 23813359697 23813359613] .
[23813359667 23813359673 23813359751]
PROG
(PARI) A320873_row(n)=vecextract(n=MagicPrimes(3*A166113[n], 3), [2, 6+n=n[2]*2==n[1]+n[3], 7-n, 9, 5, 1, 3+n, 4-n, 8]) \\ For MagicPrimes() see A073519 (the set of primes of the first row).
/* the following allows the production of all 8 variants of a magic square that are equivalent modulo reflection on any of the 4 symmetry axes of the 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
ALL(M, C(f, L)=concat(apply(f, L), L))=Set(C(REV, C(FLIP, [M, 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.
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
M. F. Hasler, Oct 22 2018
STATUS
approved