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Full Subsets of $\mathbb{N}$ Lila Naranjani
Islamic Azad University
Dolatabad Branch
Isfahan
Iran
mailto:lnaranjani@yahoo.comlnaranjani@yahoo.com

Madjid Mirzavaziri
Department of Pure Mathematics
Ferdowsi University of Mashhad
P. O. Box 1159-91775
Iran
mailto:mirzavaziri@gmail.commirzavaziri@gmail.com

Abstract:

Let $A$ be a subset of $\mathbb{N}$. We say that $A$ is $m$-full if $\sum A=[m]$ for a positive integer $m$, where $\sum A$ is the set of all positive integers which are a sum of distinct elements of $A$ and $[m]=\{1,\ldots,m\}$. In this paper, we show that a set $A=\{a_1,\ldots,a_k\}$ with $a_1<\cdots<a_k$ is full if and only if $a_1=1$ and $a_i\leq a_1+\cdots+a_{i-1}+1$ for each $i, 2\leq i\leq k$. We also prove that for each positive integer $m\notin\{2,4,5,8,9\}$ there is an $m$-full set. We determine the numbers $\alpha(m)=\min\{\vert A\vert: \sum A=[m]\}, \beta(m)=\max\{\vert A\vert: \sum A=[m]\}, L(m)=\min\{\max A: \sum A=[m]\}$ and $U(m)=\max\{\max A: \sum A=[m]\}$ in terms of $m$. We also give a formula for $F(m)$, the number of $m$-full sets.