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\begin{abstract}
For a given base $a$,  and for all integers $k$, we consider the sets
$$G_{a}(k)=\{a^{k},a^{k}+a^{k-1},\ldots,a^{k}+a^{k-1}+\dots+a^{1}+a^{0}\},$$
and for each $G_a(k)$  the corresponding ``Frobenius
set''  
$$F_a(k)=\{n \in\mathbb{N} \ |\ n \text{ is not a sum of  elements of }G_{a}(k) \}.$$
The sets $F_a(k)$ are nested and
their union is $\mathbb{N}$.   Given an integer $n$, we find the smallest $k$ such that $n \in F_a(k)$.
\end{abstract}

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