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).

1, 2, 3, 12, 15, 17, 22, 35, 124, 191, 774, 1405, 1522, 3988, 6220, 7448, 8038, 11404, 63027, 161153, 582096

Offset: 1

Author

Max Alekseyev, Feb 09 2010

Keywords

• nonn
• more

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.

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)

No other terms below 3*10^6. - Max Alekseyev, Nov 07 2024

Examples

			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)
		

Links

• Stefano Tognon, Prime Magic Squares.
• Eric Weisstein's World of Mathematics, Prime Magic Square.

Crossrefs

Cf. A064013, A073502, A073520, A164843.

Extensions

a(20) from Donovan Johnson, Nov 30 2010

a(1)=1 prepended by A.H.M. Smeets, Mar 10 2021

a(21) from Max Alekseyev, Nov 01 2024