A Note on Generalized Repunit Numerical Semigroups

Abstract

Let $A=(a_1,a_2,...,a_n)$ be relative prime positive integers with $a_i\ge 2$. The Frobenius number $F(A)$ is the largest integer not belonging to the numerical semigroup $⟨A⟩$ generated by $A$. The genus $g(A)$ is the number of positive integer elements that are not in $⟨A⟩$. The Frobenius problem is to find $F(A)$ and $g(A)$ for a given sequence $A$. In this note, we study the Frobenius problem of $A=\left(a,ba+d,b^{2}a+\frac{b^2-1}{b-1}d,...,b^{k}a+\frac{b^k-1}{b-1}d\right)$ and obtain formulas for $F(A)$ and $g(A)$ when $a\ge k-1$. Our formulas simplifies further for some special cases, such as repunit, Mersenne and Thabit numerical semigroups. The idea is similar to that in [\cite{LiuXin23},arXiv:2306.03459].

arXiv admin note: substantial text overlap with arXiv:2306.03459