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On Free Numerical Semigroups and the Construction of Minimal Telescopic Sequences
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Caleb M. Shor

Department of Mathematics

Western New England University

Springfield, MA 01119

USA

**Abstract:**

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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