Tight universal triangular forms

Abstract

For a subset $S$ of nonnegative integers and a vector $\mathbf{a}=(a_1,\dots ,a_k)$ of positive integers, let $V_S^{\prime }(\mathbf{a})=\{a_1{s}_1+\cdots +a_k{s}_k:s_i\in S\}\setminus \{0\}$. For a positive integer $n$, let $\mathcal{T}(n)$ be the set of integers greater than or equal to $n$. In this paper, we consider the problem of finding all vectors $\mathbf{a}$ satisfying $V_S^{\prime }(\mathbf{a})=\mathcal{T}(n)$, when $S$ is the set of (generalized) $m$-gonal numbers and $n$ is a positive integer. In particular, we completely resolve the case when $S$ is the set of triangular numbers.