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

Abstract

Let $A=(a_1,a_2,\dots ,a_n)$ be a sequence of 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 not in $⟨A⟩$. 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 $A=\left(a,h_{1}a+b_{1}d,h_{2}a+b_{2}d,\dots ,h_{k}a+b_{k}d\right)$ with some restrictions. An innovation is that $d$ can be a negative integer. In particular, when $A=\left(a,ba+d,b^{2}a+\frac{b^2-1}{b-1}d,\dots ,b^{k}a+\frac{b^k-1}{b-1}d\right)$, we obtain formulas for $F(A)$ and $g(A)$ when $a\ge k-1-\frac{d-1}{b-1}$. 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.

23 pages, title changed

References in corpus (1)

• A Combinatorial Approach to Frobenius Numbers of Some Special Sequences (Complete Version)