Critical numbers of natural intervals

JOURNAL OF NUMBER THEORY

卷:138

Article

Herzog, Marcel¹  Kaplan, Gil²  Lev, Arieh² 

[1] Tel Aviv Univ, Raymond & Beverly Sackler Fac Exact Sci, Sch Math Sci, IL-69978 Tel Aviv, Israel

[2] Acad Coll Tel Aviv Yafo, Sch Comp Sci, IL-61083 Tel Aviv, Israel

关键词: Natural intervals;    Sequences of integers;    Critical numbers;   

DOI  :  10.1016/j.jnt.2013.11.015

来源: Elsevier

【 摘 要 】

In this paper we study the critical numbers cr(r, n) of natural intervals [r, n]. The critical number cr(1,n) is the smallest integer l satisfying the following conditions: (i) every sequence of integers S = {r(1) = 1 <= r(2) <= ... <= r(k)} with r(1) + ... + r(k), = n and k >= l has the following property: every integer between 1 and n can be written as a sum of distinct elements of S, and (ii) there exists S with k = l, satisfying this property. The definition of cr(r, n) for r > 1 is a natural extension of cr(1, n). We completely determined the values of cr(1, n) and cr(2, n). For r > 2, we determined the values of cr(r,n) for n > 3r(2). Similar problems concerning subsets of finite groups were introduced by Erdos and Heilbronn in 1964 and extended by other authors. (C) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

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