  Let $\mathbb{N}$ denote the set of all nonnegative integers. Let $k\ge 3$ be
an integer and $A_{0} = \{a_{1}, \dots{}, a_{t}\}$ $(a_{1} < \ldots< a_{t})$ be
a nonnegative set which does not contain an arithmetic progression of length
$k$. We denote $A = \{a_{1}, a_{2}, \dots{}\}$ defined by the following greedy
algorithm: if $l \ge t$ and $a_{1}, \dots{}, a_{l}$ have already been defined,
then $a_{l+1}$ is the smallest integer $a > a_{l}$ such that $\{a_{1}, \dots{},
a_{l}\} \cup \{a\}$ also does not contain a $k$-term arithmetic progression.
This sequence $A$ is called the Stanley sequence of order $k$ generated by
$A_{0}$. In this paper, we prove some results about various generalizations of
the Stanley sequence.

Number Theory