$e^{\pi\sqrt N}$ is very close to an integer for some smallish $N$s. What about $\pi^{e\sqrt N}$? Heegner numbers (1, 2, 3, 7, 11, 19, 43, 67, 163 - let's use symbol $H_n$) are know for peculiar property that $e^{\pi\sqrt{H_n}}$ are almost integers:
$$e^{\pi \sqrt{19}}  \approx 96^3+744-0.22$$
$$e^{\pi \sqrt{43}}  \approx 960^3+744-0.00022$$
$$e^{\pi \sqrt{67}}  \approx 5280^3+744-0.0000013$$
$$e^{\pi \sqrt{163}} \approx 640320^3+744-0.00000000000075$$
(given that they are all less than 200, it goes far beyond "chance" and "randomness")
Even stranger, related to the above:
$$19 = 3 \cdot 2 \cdot 3+1$$
$$43 = 7 \cdot 2 \cdot 3+1$$
$$67 = 11 \cdot 2 \cdot 3+1$$
$$163 = 27 \cdot 2 \cdot 3+1$$
and
$$96^3 =(2^5 \cdot 3)^3$$
$$960^3=(2^6 \cdot 3 \cdot 5)^3$$
$$5280^3=(2^5 \cdot 3 \cdot 5 \cdot 11)^3$$
$$640320^3=(2^6 \cdot 3 \cdot 5 \cdot 23 \cdot 29)^3$$

Related to this, I would like to know:
Are there natural numbers N (of fairly similar range, so, lets say under 500) that produce "almost" integers in the expression $\pi^{e\sqrt{N}}$?
If yes, do they have other interesting properties, like Heegner numbers do?
If not, all right, one more reason to appreciate Heegners. :)

Up to 100000, the 10 best $N$ such that $e^{\pi\sqrt{N}}$ is almost an integer. The error $\delta$ is given such that the nearest integer is at $10^{\delta}$ from the result.
$$
\begin{array}{c|c}
N & \delta \\\hline
163 & -12.12\\
4\cdot163 & -9.79\\
9\cdot163 & -8.01\\
58 & -6.75\\
16\cdot163  & -6.51\\
67 & -5.87\\
22905 & -5.61\\
95041 & -5.55\\
54295 & -5.37\\
25\cdot163  & -5.2\\
\end{array}
$$
As you can see, no $N$ beats 163 up to 100000. (For N = 4 x 163.)
For $\pi^{e\sqrt{N}}$, the behaviour is much more regular and you obtain :
$$
\begin{array}{c|c}
N & \delta \\\hline
66972 & -5.03 \\
85516 & -5.01 \\
53204 & -4.91 \\
46665 & -4.9 \\
50237 & -4.8 \\
93909 & -4.53 \\
52970 & -4.4 \\
10024 & -4.32 \\
84702 & -4.17 \\
6814 & -4.17 \\
\end{array}$$
So, it seems there is something strange in $e^{\pi\sqrt{N}}$ that makes that question interesting !
