If a circle is centered on the origin and has radius r, it may pass through some points with integer coordinates. In fact, by symmetry, this will be a multiple of 4.
Find the smallest radius that will pass through 4n integer points where n=1,2,3,4,5.
Feel free to go further.
Any radius that goes through exactly 4n points where n is odd must have 4 of the points on either the axes or the diagonals (i.e., y= +/-x). All other points are part of a set of 8 symmetry-siblings.
Thus any radius that goes through exactly 4n points where n is odd must either be an integer or an integral multiple of sqrt(2).