A Note on Generalized Repunit Numerical Semigroups

2023/06/19 by Liu, Feihu, Xin, Guoce, Ye, Suting +1
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2306.10738

Abstract

Let A=(a₁, a₂, ..., a_n) be relative prime positive integers with a_i≥ 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=(a,ba+d,b²a+(b²-1)/(b-1)d,...,b^ka+(b^k-1)/(b-1)d) and obtain formulas for F(A) and g(A) when a≥ 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 [\citeLiuXin23,arXiv:2306.03459].

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