Full Subsets of ℕ

Lila Naranjani Islamic Azad University Dolatabad Branch Isfahan Iran

                 lnaranjani@yahoo.com

Madjid Mirzavaziri Department of Pure Mathematics Ferdowsi University of Mashhad P. O. Box 1159–91775 Iran

                 mirzavaziri@gmail.com

Abstract

Let A be a subset of ℕ. We say that A is m-full if ∑ A = [m] for a positive integer m, where ∑ A is the set of all positive integers which are a sum of distinct elements of A and [m] = {1, …, m}. In this paper, we show that a set A = {a_{1}, …, a_{k}} with a_{1} < ⋯ < a_{k} is full if and only if a_{1} = 1 and a_{i} ≤ a_{1}+⋯+a_{i−1}+1 for each i, 2 ≤ i ≤ k. We also prove that for each positive integer m ∉ {2, 4, 5, 8, 9} there is an m-full set. We determine the numbers α(m) = min{|A| : ∑ A = [m]}, β(m) = max{|A| : ∑ A = [m]}, L(m) = min{max A : ∑ A = [m]} and U(m) = max{max A : ∑ A = [m]} in terms of m. We also give a formula for F (m), the number of m-full sets.