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A173079
Positive integers n such that the sum S of 1 and first n^2-1 odd primes is divisible by n and S/n == n (mod 2).
2
1, 2, 3, 12, 15, 17, 22, 35, 124, 191, 774, 1405, 1522, 3988, 6220, 7448, 8038, 11404, 63027, 161153
OFFSET
1,2
COMMENTS
A necessary condition for the existence of n X n magic square consisting of 1 and the first n^2-1 odd primes.
In 1913, J. N. Muncey proved that 12 is actually the smallest (nontrivial) order for which such a magic square exists.
Squares of order 15, 17, 22, 35 and 124 were constructed by S. Tognon.
a(21) > 500000. - Donovan Johnson, Nov 30 2010
From A.H.M. Smeets, Mar 10 2021: (Start)
The number S/n, if it exists, is also called the potential magic constant.
It is believed that the corresponding magic squares do exist for any order a(n) with n >= 4. (End)
LINKS
Stefano Tognon, Prime Magic Squares.
Eric Weisstein's World of Mathematics, Prime Magic Square.
EXAMPLE
From A.H.M. Smeets, Mar 10 2021: (Start)
The case a(1) = 1 is trivial.
In case a(2) = 2, the set of potential magic square numbers is {1, 3, 5, 7} with potential magic constant 8, however, no magic square exists of order 2.
In case a(4) = 12, not only the potential magic constant exists, but also the magic square itself, as shown by Stefano Tognon or Eric Weisstein's World of Mathematics. (End)
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Max Alekseyev, Feb 09 2010
EXTENSIONS
a(20) from Donovan Johnson, Nov 30 2010
a(1)=1 prepended by A.H.M. Smeets, Mar 10 2021
STATUS
approved