Full subsets of \(\mathbb N\)

Naranjani, Lila ; Mirzavaziri, Madjid

Journal of integer sequences, Tome 14 (2011) no. 5

Résumé

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$ 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.

Détail

Classification : 05A17, 11P81
Keywords: perfect number, m-full set, partition of a positive integer