The Frobenius problem for generalized repunit numerical semigroups

2021/12/02 by M. B. Branco, Branco, Manuel B., Isabel Colaço +3
Mathematics · #20M14 #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Graph theory and applications #Primary: 13P10 #Rings, Modules, and Algebras #secondary: 52B20

paper · pdf · doi:10.48550/arxiv.2112.01106

openalex publication_date 2021/12/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we introduce and study the numerical semigroups generated by \a₁, a₂, … \ ⊂ ℕ such that a₁ is the repunit number in base b > 1 of length n > 1 and a_i - a_i-1 = a b^i-2, for every i ≥ 2, where a is a positive integer relatively prime with a₁. These numerical semigroups generalize the repunit numerical semigroups among many others. We show that they have interesting properties such as being homogeneous and Wilf. Moreover, we solve the Frobenius problem for this family, by giving a closed formula for the Frobenius number in terms of a, b and n, and compute other usual invariants such as the Apéry sets, the genus or the type.

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