Abstract
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 $S=\{r_1=r\leq r_2\leq \dotsb\leq r_k\}$ with $r_1+r_2 +\dotsb +r_k=n$ and $k\geq t$ 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 $r\mid n$.
Highlights
Let r, n be positive integers satisfying r n and let [r, n] denote the closed interval of integers between r and n
We begin this paper with four definitions from [13], the last of which is the definition of cr(r, n)
When we look at the cyclic group G of order n, written additively as (Zn, +), it is observed that the analogy between cr(G) and cr(1, n) is “relatively weak”, due to different restrictions on the corresponding “Spanning sets”. i.e., in the case of cr(1, n) we consider: 1. A spanning sequence (and not just a set as in the case for cr(G))
Summary
Let r, n be positive integers satisfying r n and let [r, n] denote the closed interval of integers between r and n. Every r-MZS-sequence A of G with |A| t spans Hr. We remark that parallel problems of spanning a subgroup of finite groups G (not necessarily abelian) by subsets of G were investigated in various papers (see, for example, [1, 2, 14] and [15] for details and further references). We remark that parallel problems of spanning a subgroup of finite groups G (not necessarily abelian) by subsets of G were investigated in various papers (see, for example, [1, 2, 14] and [15] for details and further references) This provides us a motivation to consider and investigate the notion of rcr(r, n) which corresponds to the spanning of the set r n = {sr|s ∈ N and sr n} (for r | n) by (r, n)-sequences (for the full definition, see Definition 12 below).
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