

A173079


Positive integers n such that the sum S of 1 and first n^21 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
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OFFSET

1,2


COMMENTS

A necessary condition for the existence of n X n magic square consisting of 1 and the first n^21 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.
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



EXAMPLE

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



EXTENSIONS



STATUS

approved



