The Frobenius Number of Geometric Sequences (original) (raw)

2015

The Frobenius problem is about finding the largest integer that is not contained in the numerical semigroup generated by a given set of positive integers. In this paper, we derive a solution to the Frobenius problem for sets of the form {mk,mk−1n,mk−2n2,..., nk}, where m,n are relatively prime positive integers. 1.

Numerical semigroups with a given set of pseudo-Frobenius numbers

LMS Journal of Computation and Mathematics, 2016

The pseudo-Frobenius numbers of a numerical semigroup are those gaps of the numerical semigroup that are maximal for the partial order induced by the semigroup. We present a procedure to detect if a given set of integers is the set of pseudo-Frobenius numbers of a numerical semigroup and, if so, to compute the set of all numerical semigroups having this set as set of pseudo-Frobenius numbers.

Associated semigroups of numerical sets with fixed Frobenius number

arXiv: Combinatorics, 2019

A numerical set is a co-finite Subset of the natural numbers that contains zero. Its Frobenius number is the largest number in its compliment. A numerical semigroup is a numerical set that is closed under addition. Each numerical set has an associated semigroup A(T)=t∣t+TsubseteqTA(T)=\{t|t+T\subseteq T\}A(T)=t∣t+TsubseteqT, which is a numerical semigroup with the same Frobenius number as that of TTT. For a fixed Frobenius number fff there are 2f−12^{f-1}2f−1 numerical sets. We give a complete asymptotic description of what percentage of these numerical sets are mapped to which semigroups. We also obtain parallel results for symmetric numerical sets.

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