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

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

Author

M. F. Hasler, Oct 22 2018

Keywords

• nonn
• fini
• full

Comments

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).

Examples

			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]
		

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

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

Crossrefs

Cf. A073519, A073520, A073521, A073522.

Programs

PARI

/* 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.