We study a hierarchy $\{\mathcal{L}_2^k \}$ of Kalmàr elementary functions on integers based on a classification of LOOP programs of limited complexity, namely those in which the depth of nestings of LOOP commands does not exceed two. It is proved that $n$-place functions in $\mathcal{L}_2^k$ can be enumerated by a single function in $\mathcal{L}_2^{k+2}$, and that the resulting hierarchy of elementary predicates (i.e., functions with 0,1-values) is proper in that there are $\mathcal{L}_2^{k+2}$ predicates that are not in $\mathcal{L}_2^{k}$. Along the way the rudimentary predicates of Smullyan are classified as $\mathcal{L}_2^{2}$.