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}, …, a^{k} + a^{k−1} + ⋯ + a^{1} + a^{0}}, and for each G_{a}(k) the corresponding “Frobenius set”

F_{a}(k) = {n ∈ ℕ | n is not a sum of elements of G_{a}(k)}. The sets F_{a}(k) are nested and their union is ℕ. Given an integer n, we find the smallest k such that n ∈ F_{a}(k).