A diophantine equation in $\mathbb{N}$

This is an elaboration of Emil Jeřábek's important comment, and contains no original contribution. The OP's problem was examined in depth by Borwein-Choi (1999), and their article is available for free here. I will summarize the content of this article below.

Let us consider an integer $n\geq 2$ that cannot be written as $ab+bc+ca$ with integers $a,b,c\geq 1$. Using Lemma 1.1 with $k=1$, we see that $n$ is even. Using Theorem 2.6, it follows that either $n\in\{4,18\}$ or $n=2p_1\dots p_r$ with distinct odd primes $p_j$. The proof of these two results are elementary but highly nontrivial. Then, using some results of Andrews and Crandall, the authors deduce that the class number $h(-4n)$ equals $2^r$ (which is the number of genera of discriminant $-4n$). It is classical that $h(-4n)$ is of size $n^{1/2+o(1)}$ while $2^r=n^{o(1)}$, hence the list of exceptional $n$'s is certainly finite. In fact, Weinberger (1973) analyzed the condition $h(-4n)=2^r$ carefully, and this way we know that either $$n\in\{2,4,6,10,18,22,30,42,58,70,78,102,130,190,210,330,462\},$$ or $n$ is possibly a further single number beyond $10^{11}$. The last possibility can only occur if the Generalized Riemann Hypothesis (GRH) fails for the $L$-function of some quadratic Dirichlet character.

To summarize, there are $18$ exceptional integers $n\geq 1$ if GRH holds, and possibly one further exceptional $n\geq 1$ if GRH fails. (In the above paragraph I restricted to $n\geq 2$ for convenience.)