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

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

Table of n, a(n) for n=1..20.

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

Cf. A064013, A073502, A073520, A164843.

Sequence in context: A163906 A302844 A180630 * A173903 A154785 A090512

Adjacent sequences: A173076 A173077 A173078 * A173080 A173081 A173082

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