A232803 Odd primes, twice odd primes, 4, and 8.

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

Offset: 1

Author

J. Lowell, Nov 30 2013

Keywords

• nonn

Comments

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

Examples

			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).
		

Links

• Allan Adler, Math Forum: Magic Squares: 'Multiplication' in a new context
• Allan Adler, Math Forum: Proof that there are no 2x2 magic squares
• Allan Adler, Math Forum: Multiplying Two Magic Squares
• Allan Adler, Math Forum: 12x12 Magic Square

Programs

PARI

isok(n) = (isprime(n) && (n%2)) || (n==8) || (!(n%2) && isprime(n/2)); \\ Michel Marcus, Dec 07 2013

Extensions

More terms from Michel Marcus, Dec 07 2013

Replaced definition with a more explicit one, following the comments of Michel Marcus. - N. J. A. Sloane, Dec 19 2019