Heterosquares and less restrictive variations

[Leroy Quet]

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