The Frobenius Problem for the Proth Numbers

Let n be a positive integer greater than 2. We define 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.