The Frobenius Problem for the Proth Numbers

Abstract

Let $n$ be a positive integer greater than $2$. We define \textit{the Proth numerical semigroup}, $P_{k}(n)$, generated by $\{k 2^{n+i}+1 \,\mid\, i \in \mathbb{N}\}$, where $k$ is an odd positive number and $k < 2^{n}$. In this paper, we introduce the Frobenius problem for the Proth numerical semigroup $P_{k}(n)$ and give formulas for the embedding dimension of $P_{k}(n)$. We solve the Frobenius problem for $P_{k}(n)$ by giving a closed formula for the Frobenius number. Moreover, we show that $P_{k}(n)$ has an interesting property such as being Wilf.