Journal of Integer Sequences, Vol. 22 (2019), Article 19.2.4

On Free Numerical Semigroups and the Construction of Minimal Telescopic Sequences

Caleb M. Shor
Department of Mathematics
Western New England University
Springfield, MA 01119


A free numerical semigroup is a submonoid of the non-negative integers with finite complement that is additively generated by the terms in a telescopic sequence with gcd 1. However, such a sequence need not be minimal, which is to say that some proper subsequence may generate the same numerical semigroup, and that subsequence need not be telescopic. In this paper, we will see that for a telescopic sequence with any gcd, there is a minimal telescopic sequence that generates the same submonoid. In particular, given a free numerical semigroup we can construct a telescopic generating sequence that is minimal. In the process, we will examine some operations on and constructions of telescopic sequences in general.

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Received June 27 2018; revised versions received January 2 2019; January 6 2019. Published in Journal of Integer Sequences, February 23 2019.

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