On the Frobenius Number and Genus of a Collection of Semigroups Generalizing Repunit Numerical Semigroups

Abstract

Let $A=(a_1, a_2, \ldots, a_n)$ be a sequence of relative prime positive integers with $a_i\geq 2$. The Frobenius number $F(A)$ is the largest integer not belonging to the numerical semigroup $\langle A\rangle$ generated by $A$. The genus $g(A)$ is the number of positive integer elements not in $\langle A\rangle$. The Frobenius problem is to determine $F(A)$ and $g(A)$ for a given sequence $A$. In this paper, we study the Frobenius problem of $A=\left(a,h_1a+b_1d,h_2a+b_2d,\ldots,h_ka+b_kd\right)$ with some restrictions. An innovation is that $d$ can be a negative integer. In particular, when $A=\left(a,ba+d,b^2a+\frac{b^2-1}{b-1}d,\ldots,b^ka+\frac{b^k-1}{b-1}d\right)$, we obtain formulas for $F(A)$ and $g(A)$ when $a\geq k-1-\frac{d-1}{b-1}$. Our formulas simplify further for some special cases, such as Mersenne, Thabit, and repunit numerical semigroups. Finally, we partially solve an open problem for the Proth numerical semigroup.