Let $A=\{mn(m+n)\mid n,m\in \mathbb{N}_0\}$. Sorted, this is OEIS sequence [A088915][1]. 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 fully reducible cubic polynomials $P,P+a \in \mathbb{Z}[X]$ if and only if $a\in AA$. *Fully reducible* means that $P$ and $P+a$ are both products of 3 linear terms.

  [1]: https://oeis.org/search?q=2%2C6%2C12%2C16%2C20%2C30%2C42%2C48&language=english&go=Search