Schinzel's Theorem states that there are a set of circles with a given number of integer points on the circumference of the circle. The theorem includes the equation for an instance of circle given $n$.

However, the formula does not always give the circle with minimum radius. I am looking for a formula, or at least a procedure to generate the circle with minimum radius given $n$. A brute force search gives first values of the series.

\begin{array}{ccc} \text{center} & \text{square of radius} & \text{number of points on c} \\ \left(\frac{1}{2},\frac{1}{2}\right) & \frac{1}{2} & 4 \\ \left(\frac{1}{6},\frac{1}{6}\right) & \frac{625}{18} & 5 \\ \left(\frac{1}{2},0\right) & \frac{25}{4} & 6 \\ \left(\frac{9}{22},\frac{5}{22}\right) & \frac{138125}{242} & 7 \\ \left(\frac{1}{2},\frac{1}{2}\right) & \frac{5}{2} & 8 \\ \left(\frac{1}{6},\frac{1}{6}\right) & \frac{4225}{18} & 9 \\ \left(\frac{1}{2},0\right) & \frac{625}{4} & 10 \\ \end{array}

I asked the question on math exchange site but got nothing better than brute force searching. I am wondering if there is formulas or existing research and paper on the problem.

Schinzel's Theorem states that there are a set of circles with a given number of integer points on the circumference of the circle. The theorem includes the equation for an instance of circle given $n$.

However, the formula does not always give the circle with minimum radius. I am looking for a formula, or at least a procedure to generate the circle with minimum radius given $n$. A brute force search gives first values of the series.

\begin{array}{ccc} \text{center} & \text{square of radius} & \text{number of points on c} \\ \left(\frac{1}{2},\frac{1}{2}\right) & \frac{1}{2} & 4 \\ \left(\frac{1}{6},\frac{1}{6}\right) & \frac{625}{18} & 5 \\ \left(\frac{1}{2},0\right) & \frac{25}{4} & 6 \\ \left(\frac{9}{22},\frac{5}{22}\right) & \frac{138125}{242} & 7 \\ \left(\frac{1}{2},\frac{1}{2}\right) & \frac{5}{2} & 8 \\ \left(\frac{1}{6},\frac{1}{6}\right) & \frac{4225}{18} & 9 \\ \left(\frac{1}{2},0\right) & \frac{625}{4} & 10 \\ \end{array}

There is a similar question on math exchange site but the answer is still brute force searching. I am wondering if there is formulas or existing research and paper on the problem.