Suppose I have a positive number $d \in \mathbb{R}$ and a sequence of numbers $a\_n \in [0,d]$ for $n \in \mathbb{N}$ with the following properties 
$$
\sum\_{i=1}^{\infty} a\_i^r \in \mathbb{Z}
$$
for all $r \in \mathbb{N}$ and 
$$
\sum\_{i=1}^{\infty} a\_i \leq d \ .
$$



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Note that it does not follow that $a\_i \in \mathbb{Z}$ as the sequence $a\_1 = 2 + \sqrt{2}$, $a\_2 = 2 - \sqrt{2}$, $a\_k = 0$ for $k > 2$ with $d=4$ shows. This sequence also shows that the power sums can be unbounded.



Integer-valued power sums
The function 
$$
f : z \in \mathbb{C} \longmapsto \sum\_{i} \frac{a\_i}{1-a\_iz}
$$
is meromorphic on $\mathbb{C}$ and has integral Taylor coefficients. It follows from a theorem of Borel that such a function must be in $\mathbb{Q}(z)$; see for example Richard Stanley's answer [here](https://mathoverflow.net/questions/6179/does-there-exist-a-meromorphic-function-all-of-whose-taylor-coefficients-are-pri). In particular $f$ has only finitely many poles ; this implies that only finitely many of the $a\_i$'s are nonzero.


