# Heterosquares and...Prime Squares

zak seidov zakseidov at yahoo.com
Wed Aug 20 13:05:59 CEST 2008

```There are exactly 1152 3x3 squares filled with digits 1..9,
such that each row and column has prime sum.

Lexicographically the first such square is

124
586
739

with row sums 7,19,19, and column sums 13,13,19.

Lexicographically the last such square is

986
524
371

with row sums 23,11,11, and column sums 17,17,11.

From these 1152 squares,
no one has additionally both main diagonals with prime sum.

Is it all old hat?

Thanks, zak

--- On Tue, 8/19/08, Leroy Quet <q1qq2qqq3qqqq at yahoo.com> wrote:

> From: Leroy Quet <q1qq2qqq3qqqq at yahoo.com>
> Subject: Heterosquares and less restrictive variations
> To: seqfan at ext.jussieu.fr
> Cc: qq-quet at mindspring.com
> Date: Tuesday, August 19, 2008, 7:09 PM
> A "heterosquare" is like a magic-square, except
> that we want the the sums of the columns, of the rows, and
> of the main diagonals to each be DISTINCT. The integers
> 1,2,3,...,n^2 are put in the cells of the square, of course.
>
>
> http://mathworld.wolfram.com/Heterosquare.html
>
> There is a sequence in the EIS of the number of
> "antimagic" squares modulo reflections and
> rotations, sequence A050257. (An antimagic square is a more
> restrictive version of a heterosquare, where the sums form a
> sequence of consecutive integers.)
>
> http://mathworld.wolfram.com/AntimagicSquare.html
>
>
> But doing a search for "heterosquare" or
> "hetero-square" on the EIS brings up no hits.
>
> Could the numbers of such squares be in the EIS under a
> different name?
>
> Then we could also consider the heterosquares with the
> relaxed condition that the diagonal sums don't count.
> (There is the sequence of the numbers of such squares
> modulo rotations and reflections, and the sequence where
> each rotation and reflection is counted separately.)
> Are either of these sequences in the EIS?
>
> Then we can relax the condition that the rows add up to
> distinct values, and consider only the squares where the
> columns all add up to distinct values.
>
> Now the sequence of the numbers of such squares MUST be in
> the EIS. Or is it?
>
>
> Thanks,
> Leroy Quet

```