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For a given base $ a$, and for all integers $ k$, we consider the sets

$\displaystyle G_{a}(k)=\{a^{k},a^{k}+a^{k-1},\ldots,a^{k}+a^{k-1}+\dots+a^{1}+a^{0}\},$

and for each $ G_a(k)$ the corresponding ``Frobenius set''

$\displaystyle F_a(k)=\{n \in\mathbb{N} \ \vert\ n$    is not a sum of elements of $\displaystyle G_{a}(k) \}.$

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)$.

Jeffrey Shallit 2008-12-14