S-packing colorings of distance graphs with distance sets of cardinality $2$

For a non-decreasing sequence S=(s₁,s₂,…) of positive integers, a partition of the vertex set of a graph G into subsets X₁,…, X_, such that vertices in Xᵢ are pairwise at distance greater than sᵢ for every i∈\1,…,\, is called an S-packing -coloring of G. The minimum for which G admits an S-packing -coloring is called the S-packing chromatic number of G, denoted by χS(G). In this paper, we consider S-packing colorings of distance graphs G(Z,,t\), where k and t are positive integers, which are the graphs whose vertex set is Z, and two vertices x,y∈ Z are adjacent whenever |x-y|∈,t\. We complement partial results from two earlier papers, thus determining all values of χS(G(Z,,t\)) when S is any sequence with sᵢ≤ 2 for all i. In particular, if S=(1,1,2,2,…), then the S-packing chromatic number is $2$ if $k+t$ is even, and $4$ otherwise, while if S=(1,2,2,…), then the S-packing chromatic number is $5$, unless ,t\=\2,3\ when it is $6$; when S=(2,2,2,…), the corresponding formula is more complex.