Journal of Integer Sequences, Vol. 12 (2009), Article 09.1.2

Characterizing Frobenius Semigroups by Filtration

Inga Johnson, Sean Powers, Colin Starr, Charles Trevelyan and Craig Webster
Department of Mathematics
Willamette University
Salem, OR 97301


For a given base $ a$, and for all integers $ k$, we consider the sets

$\displaystyle G_{a}(k)=\{a^{k},a^{k}+a^{k-1},\ldots,a^{k}+a^{k-1}+\dots+a^{1}+a^{0}\},$

and for each $ G_a(k)$ the corresponding ``Frobenius set''

$\displaystyle F_a(k)=\{n \in\mathbb{N} \ \vert\ n$    is not a sum of elements of $\displaystyle G_{a}(k) \}.$

The sets $ F_a(k)$ are nested and their union is $ \mathbb{N}$. Given an integer $ n$, we find the smallest $ k$ such that $ n \in F_a(k)$.

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(Concerned with sequence A023758.)

Received August 1 2006; revised versions received August 31 2007; September 13 2008; October 20 2008. Published in Journal of Integer Sequences, December 14 2008.

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