Generating Seeds

Let $D$ be a positive integer.

Suppose that we would like to find all nonnegative integer solutions $(x,y,z)$ of $$x^2+D=y\cdot z.$$

Let's assume that $(x,y,z)$ is a solution of the above equation. Then, it can be verified that $(x+y,y,2x+y+z)$ and $(x+z,2x+y+z,z)$ are also solutions of the equation. Let's define this generating process as the evolution of $(x,y,z)$.

Surprisingly, we can find all solutions uniquely by choosing some seeds $S_D=\{(x_1,y_1,z_1),\dots ,(x_k,y_k,z_k)\}$ and evolving them repeatedly. [a seed is a solution of the equation.]

For example, when $D=2$, we can choose $S_2$ as $S_2=\{(0,1,2),(0,2,1)\}$.

Let $C(D)$ be the minimum number of seeds needed to enumerate all nonnegative integer solutions of the equation.

It can be verified that $C(2)=2$, $C(3)=3$ and $C(100)=18$.

Let $S(n):=\sum _{D=1}^{n}C(D)$. You are given $S(10)=40$ and $S(100)=1714$.

Find $S(3\cdot {10}^7)$.

[My Timing: 14.8 seconds (PyPy)]