$$\tan{e^2}=1.99362\ldots$$ $$\tan{e^3}=2.87427\ldots$$ Why are these close to $2$ and $3$? The former seems to be especially close to $2$.
Or why are the solutions of $\tan e^x=x$ near $x\approx2$ and $x\approx3$ so close to integers?
The solutions of $\tan e^x=x$ are as follows: $$x=1.40987943\ldots\\ x=2.00017774\ldots\\ x=2.36036677\ldots\\ x=2.62270264\ldots\\ x=2.82962192\ldots\\ x=3.00065349\ldots\\ x=3.27363173\ldots\\ x=3.38635403\ldots\\ x=3.48760393\ldots\\ x=3.57950793\ldots\\ x=3.66365002\ldots\\ x=3.74124217\ldots\\ x=3.81323250\ldots\\ x=3.88037684\ldots\\ x=3.94328751\ldots\\ x=4.00246750\ldots\\ x=4.05833510\ldots\\ \vdots$$ It seems to me that the two solutions specifically close to integers. Does anyone have an explanation for this?