Full Subsets of N

Let

be a subset of $\mathbb{N}$ . We say that

-full if $\sum A=[m]$ for a positive integer

, where $\sum A$ is the set of all positive integers which are a sum of distinct elements of

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

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

-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

. We also give a formula for

, the number of

-full sets.

Full Subsets of N

Lila Naranjani Islamic Azad University Dolatabad Branch Isfahan Iran Madjid Mirzavaziri Department of Pure Mathematics Ferdowsi University of Mashhad P. O. Box 1159-91775 Iran

Lila Naranjani
Islamic Azad University
Dolatabad Branch
Isfahan
Iran

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