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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
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
Sequence in context: A163906 A302844 A180630 * A173903 A154785 A090512
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.)