Just stumbled upon $$\ln2\approx 0.4^{0.4}$$ and wondered if that's just a coincidence, or whether there's some deeper reason?
$$\ln2 - 0.4^{0.4}\approx 0.00000234$$ which is a relative error of just $3.37\cdot10^{-6}$, which is remarkable (better than 17 bits / better than 5 decimals). It's $$ 0.4^{0.4} = \sqrt[5]{4/25} $$ but I still do not see a connection. The values are: $$\begin{align} \ln2 &\approx 0.69314718 \\ 0.4^{0.4} &\approx 0.69314484 \end{align}$$