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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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified July 19 11:36 EDT 2024. Contains 374394 sequences. (Running on oeis4.)