MathNet 2017-part-07 — 172 problems (EN–VI) Question 128

A sequence of positive integers $a_{1}, a_{2}, \ldots, a_{2017}$ has the property that for all integers $m$ where $1 \leq m \leq 2017$, $3\left(\sum_{i=1}^{m} a_{i}\right)^{2}=\sum_{i=1}^{m} a_{i}^{3}$. Compute $a_{1337}$.