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''

$$F_a(k)=\{n \in\mathbb{N} \ \vert\ n$$    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 $n \in F_a(k)$.