R-critical numbers of natural intervals

The critical number cr(r, n) of natural intervals [r, n] was introduced by Herzog, Kaplan and Lev in 2014. The critical number cr(r, n) is the smallest integer t satisfying the following conditions: (i) every sequence of integers S = {r₁ = r ≤ r₂ ≤ · · · ≤ r_k} with r₁ +r₂ +· · · +r_k = n and k ≥ t has the following property: every integer between r and n − r can be written as a sum of distinct elements of S, and (ii) there exists S with k = t, which satisfies that property. In this paper we study a variation of the critical number cr(r, n) called the r-critical number rcr(r, n). We determine the value of rcr(r, n) for all r, n satisfying r | n.