Let $A=\{mn(m+n)\mid n,m\in \mathbb{N}_0\}$. Sorted, this is OEIS sequence A088915. What is its asymptotic behavior? It seems approximately $a(n)=O(n^{1.5})$, but not quite.

I'm actually even more interested in the asymptotic behavior of the sequence given by $AA$, the set of products of two elements of $A$.

Any ideas on how to approach these questions?

For background, there are monic completely split cubic polynomials $P,P+a \in \mathbb{Z}[X]$ if and only if $a\in AA$. By completely split (is that the right terminolgy?) I mean that $P$ and $P+a$ are both products of 3 linear terms.