I know from reading that the Heegner number 163 yields the prime generating or Euler Lucky Number 41. Now apparently $\ln23<\pi$ and this can be shown without calculators. I noticed that $$ \pi-\ln23= \cfrac{1}{\color{red}{163} + \cfrac{1}{1 + \cfrac{1}{\color{red}{41} + \cfrac{1}{2 + \cdots}}}} $$
Question: Are there any "good" mathematical reasons why the largest Heegner and largest Euler Lucky number occur within the first three (-four ?) terms of the expansion? Or is it purely coincidence ?
Indeed the finite c.f. $$\cfrac{1}{\color{red}{163} + \cfrac{1}{1 + \cfrac{1}{\color{red}{41}}}}:=\frac{42}{6887}\approx 0.00609843\ldots.$$ In turn this yields the crude approximation $$\ln 23+\frac{42}{6887}\approx\pi;$$ which I believe gives the first 8 digits of $\pi$ correctly.
