It can be shown that the cardinality of

$$R(X) = \{n \in \mathbb{N} : \exists u,v \in \mathbb{Z} \text{ s.t. } n = uv(u+v), n \leq X\}$$

satisfies

$$\displaystyle R(X) = C X^{2/3}(1 + o(1))$$

for some explicit constant $C$. Indeed, by the main result of this paper and the fact that $R(X)$ above is equal to half the size of the counting function in that paper, one gets that

$$\displaystyle R(X) = \frac{\Gamma^2(1/3)}{4\Gamma(2/3)} X^{2/3} + O(X^{1/2}).$$

We do not need the full strength of my paper with Cam Stewart in this case, since things are easier when the binary form is totally reducible. The totally reducible case has already been mostly done by Christopher Hooley in an earlier paper.

The counting function $R(X)$ I used is not exactly the one that you asked... since you insist that $m,n$ are positive. I suspect that a little bit more work would allow you to tweak the constant. The error term will not be problematic.

In general, it is not known whether there exist an infinite increasing sequence $\{k_\ell\}$ of natural numbers such that the sets $S_\ell = \{(m,n) \in \mathbb{Z}^2 : \gcd(m,n) = 1, mn(m+n) = k_\ell\}$ satisfy $|S_{\ell + 1}| > |S_{\ell}|$ for all $\ell \geq 1$. If such a sequence exists, then there exists a sequence of elliptic curves whose Mordell-Weil rank tends to infinity.

However, one can show (as is done in full generality in the above linked paper) that such occurrences are rare: that is, for 100% of integers representable by a non-singular integral binary form of degree $d \geq 3$, the only repetitions are accounted for by $\operatorname{GL}_2(\mathbb{Q})$-automorphisms.

with multiplicitiesof the number of terms in $a$ up to $N$, then $S\approx \frac{1}{2}\int_1^{\sqrt{N}}\big(\sqrt{n^2+\frac{4N}{n}}-n\big)dn$... (cont) $\endgroup$ – Yaakov Baruch Mar 17 at 12:56without multiplicities, $T/(2.6499581...\times N^{2/3})$ seems to be increasing to a limit $\le1$, so that likely $T=O(N^{2/3})$. The definite integral above seems to match OEIS sequence A197374. $\endgroup$ – Yaakov Baruch Mar 17 at 12:56