On the Numerical Semigroup Generated by 2 k − m ⋅ 2 n + i − 1 | i ∈ ℕ

Abstract

Let $k,n$ be two positive integers such that $2\le k\le 2^n$ and $m$ be an odd number with $1\le m<2^{k-1}$ . Denote by $T\left(k,m,n\right)$ a numerical semigroup generated by $\left\{\left(2^k-m\right)\cdot 2^{n+i}-1|i\in \mathbb{N}\right\}$ . In this paper, we give the formula for the embedding dimension of $T\left(k,m,n\right)$ . In particular, the formulas for computing the Frobenius number and the genus of $T\left(k,2^t+1,n\right)$ with $2\le t<k-1$ are given.