Any reason why these numbers are close to integers?

Question

I've heard about Heegner numbers some time ago and have been playing today with their property which seems very intriguing to me (and, to be fair, the one I can understand most easily), namely, the fact that $e^{\pi\sqrt{d}}$ turns out to be quite close to an integer when $d$ is a Heegner number. For example, somewhat most famously, $e^{\sqrt{163} \pi } =262537412640768743+0.9999999999993\ldots$

So, I've just computed the numbers $\exp(\pi d^p)$ for $p$ in $(\frac{1}{2},\frac{1}{3},\frac{2}{3},\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5})$ and $d$ from $1$ to $10000$ with this code in Mathematica (if anyone is interested):

x = 3000
DeleteCases[
  Prepend[Map[If[(# > 10^-4) && ((1 - #) > 10^-4), 0, #] &, 
    N[Table[{FractionalPart[N[Exp[\[Pi] Sqrt[i]], x]], 
       FractionalPart[N[Exp[\[Pi] i^(1/3)], x]], 
       FractionalPart[N[Exp[\[Pi] i^(2/3)], x]], 
       FractionalPart[N[Exp[\[Pi] i^(1/5)], x]], 
       FractionalPart[N[Exp[\[Pi] i^(2/5)], x]], 
       FractionalPart[N[Exp[\[Pi] i^(3/5)], x]], 
       FractionalPart[N[Exp[\[Pi] i^(4/5)], x]], i}, {i, 10000}], 
     15], {2}], {1/2, 1/3, 2/3, 1/5, 2/5, 3/5, 4/5}], {0, 0, 0, 0, 0, 
   0, 0, _?NumberQ}] // TableForm

There was some list of numbers, from which I found that $e^{\sqrt[3]{3163} \pi } = 106618076286425992470 + 0.99997\ldots\\$, $e^{\sqrt[3]{7577} \pi } = 627840988101335617898564443+0.99991\ldots\\$, $e^{\sqrt[5]{7607} \pi } = 141371353+0.000010\ldots$ and a bunch of other examples (nothing so striking as $e^{\sqrt{163} \pi }$ though).

Is there some reason for this (maybe expansion of some generalization of j-invariant) or is it just a byproduct of taking huge amount of numbers?