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A232803
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Odd primes, twice odd primes, 4, and 8.
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3
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3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 17, 19, 22, 23, 26, 29, 31, 34, 37, 38, 41, 43, 46, 47, 53, 58, 59, 61, 62, 67, 71, 73, 74, 79, 82, 83, 86, 89, 94, 97, 101, 103, 106, 107, 109, 113, 118, 122, 127, 131, 134, 137, 139, 142, 146, 149, 151, 157, 158, 163, 166
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OFFSET
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1,1
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COMMENTS
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Also numbers n for which all possible n X n magic squares are prime.
Note that there are no 2 X 2 magic squares.
All primes, except 2, belong to this sequence. This is because p X p magic squares, with p primes, cannot be derived from smaller magic squares. Otherwise p would be equal to a product of smaller integers. Also, since there are no 2 X 2 magic squares, we cannot have a (2p) X (2p) that could be derived from smaller magic squares. And also we cannot have an 8 X 8 derived magic square (see first example). So this sequence is A065091 (odd primes) U A100484 (even semiprimes) U {8}. And A100484 U {8} is also A161344 (see second comment by Zak Seidov there). So this sequence is: A065091 U A161344. - Michel Marcus, Dec 07 2013
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LINKS
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EXAMPLE
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8 qualifies because a composite 8 X 8 magic square is impossible, such a square would require a 2 X 2 magic square, and there are none (see 2nd link).
9 is not part of sequence because a 9 X 9 magic square can be created by multiplying a 3 X 3 magic square by itself.
Similarly 12 is not part of sequence because a 12 X 12 magic square can be created by multiplying a 3 X 3 magic square and a 4 X 4 magic square (see 3rd and 4th links).
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PROG
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(PARI) isok(n) = (isprime(n) && (n%2)) || (n==8) || (!(n%2) && isprime(n/2)); \\ Michel Marcus, Dec 07 2013
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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