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

TL;DR

This work addresses the Frobenius problem and genus for a broad family of numerical semigroups generalizing repunit semigroups by introducing Collection GCNS with $A=(a,Ha+dB)$. The authors reduce the computation of $F(A)$ and $g(A)$ to a minimization problem $O_B^H(M)$ and leverage greedy algorithms enabled by orderly base sequences and Apéry-set techniques to produce closed-form formulas under mild conditions. They provide explicit formulas for representative specializations, including repunit, Mersenne, Thabit, and Proth semigroups, and they partially resolve an open problem for Proth semigroups, thereby unifying several known families under a single analytical framework. This unified approach yields practical tools for exact Frobenius-type invariants across a spectrum of semigroups and clarifies connections among classical families in the literature.

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.

Theorems & Definitions (55)

• Lemma 2.1: J. E. Shockley, E. S. Selmer
• Lemma 2.2
• Proposition 2.3
• Lemma 2.4
• proof
• Lemma 2.5: One-Point Theorem, LJCowenTCHuMJMagazine
• Lemma 2.6
• proof
• Proposition 2.7
• proof