cp's OEIS Frontend

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).

%I A173079 #39 Feb 16 2025 08:33:11
%S A173079 1,2,3,12,15,17,22,35,124,191,774,1405,1522,3988,6220,7448,8038,11404,
%T A173079 63027,161153,582096
%N 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).
%C A173079 A necessary condition for the existence of n X n magic square consisting of 1 and the first n^2-1 odd primes.
%C A173079 In 1913, J. N. Muncey proved that 12 is actually the smallest (nontrivial) order for which such a magic square exists.
%C A173079 Squares of order 15, 17, 22, 35 and 124 were constructed by S. Tognon.
%C A173079 From _A.H.M. Smeets_, Mar 10 2021: (Start)
%C A173079 The number S/n, if it exists, is also called the potential magic constant.
%C A173079 It is believed that the corresponding magic squares do exist for any order a(n) with n >= 4. (End)
%C A173079 No other terms below 3*10^6. - _Max Alekseyev_, Nov 07 2024
%H A173079 Stefano Tognon, <a href="http://digilander.libero.it/ice00/magic/index.html">Prime Magic Squares</a>.
%H A173079 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeMagicSquare.html">Prime Magic Square</a>.
%e A173079 From _A.H.M. Smeets_, Mar 10 2021: (Start)
%e A173079 The case a(1) = 1 is trivial.
%e A173079 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.
%e A173079 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)
%Y A173079 Cf. A064013, A073502, A073520, A164843.
%K A173079 nonn,more
%O A173079 1,2
%A A173079 _Max Alekseyev_, Feb 09 2010
%E A173079 a(20) from _Donovan Johnson_, Nov 30 2010
%E A173079 a(1)=1 prepended by _A.H.M. Smeets_, Mar 10 2021
%E A173079 a(21) from _Max Alekseyev_, Nov 01 2024