One, if it is legal, many otherwise

Seriously: Is there a sudoku with exactly 2006 solutions?

Or must the number of solutions be even, of have

special factors?

I cannot imagine an answer myself, and my solver

does not do brute force yet.

Anyway, I assume the "even" hypothesis is surely wrong, as ruuds

benchmark list has a sudoku with 3673 solutions, which isn't even

divisible by three.

Just curious, any clues?