$\displaystyle 1663 \approx frac{1663e^2}{3} \frac{3\times 2^{12}}{e^2}$approx 2^{12}$.
showed up in a comment of this mostly-unrelated question.
Numerically, it's not a surprise that $e^2$ is close to a rational number whose numerator and denominator are in this range — similarly good approximations to most numbers can be obtained by truncating the continued fraction at the desired level of accuracy. What's much more of a surprise to me is the appearance of a 12th power in this expression. Is there a good explanation for this, e.ghaving such a smooth number here? E.g. like see the j-invariant explanation for Ramanujan's observation that
$e^{\pi\sqrt{163}}$
is close to an integer, or like the Pisot number explanation for the fact that even powers of the golden ratio are close to integers?.
