Can you find the smallest radius sphere that has points with integer values?

Stepping up from this.

How about if we take a sphere instead of a circle. For example, a sphere of radius sqrt(38) will have integer coordinates here:

2,3,5 (and -2,3,5 2,-3,5, 2,3,-5, -2,-3,5 -2,3,-5 -,2,-3,5 -2,3,5)
2,5,3 (and 7 more)
3,2,5 (and 7 more)
3,5,2 (and 7 more)
5,2,3 (and 7 more)
5,3,2 (and 7 more)

6,1,1 (3 permutations * 2^3 arrangements of negative signs = 24 total)

for a total of 72.

Triples of the form x^2 +y^2 + z^2 = r^2 will generate
42 points if x,y,and z are unique
or 24 points if there are two unique numbers
or 8 it they are all the same.

Is 7 the lowest radius for 54 points? Is it the smallest radius for an integer sphere with more than 15 contact points?

Is sqrt(38) the lowest radius for 72 points?

Is the sphere with the most integer intersects always centered on the origin?

What are some other minimum radii for integer coordinate points on the sphere?

sqrt(86) has 108 points. Is that the minimum?
sqrt(101) has 144 points. Is that the minimum?

What’s the minimum integer radius for a sphere with more than 95 points?