Square + 1 = Squarefree

Let $C (n)$ be the number of squarefree integers of the form $x^{2} + 1$ such that $1 \leq x \leq n$ .

For example, $C (10) = 9$ and $C (1000) = 895$ .

Find $C (123567101113)$ .

Write-up coming later

The complete problem is available here. An approach, code, and answer will be added later.