It is relatively easy to prove that there exist infinitely many positive integers not of the form $2ij+i+j$, where $i,j\in \mathbb N$.
Sundaram already found that some positive integer is prime if and only if it is of the form $2k+1$, were $k$ is some positive integer not of the form $2ij+i+j$. The proof is pretty straightforward, as $2(2ij+i+j)+1=(2i+1)(2j+1)$. And as we have many proofs of the existence of infinitely many prime numbers, we can use any of them together with Sundaram's result to prove the existence of infinitely many positive integers not of the form $2ij+i+j$.
My question
Do you know of / can you share any proof of the existence of infinitely many positive integers not of the form $2ij+i+j$ that does not use neither Sundaram's result nor some classical proofs of the existence of infinitely many prime numbers? For instance, a proof using sieve theory would be acceptable.
Thanks in advance!
EDIT
As a clarification, I am not looking for alternative proofs that there are infinitely many primes. I am looking for a proof that there are infinitely many positive integers not of the form $2ij+i+j$, because maybe it can be useful to derive insights for the generalized form $Kij+i+j$, $K\in\mathbb N$. Incidentally, the fact that there are infinitely many primes tell us (through Sundaram's) that there are infinitely many positive integers not of the form $2ij+i+j$, but I have not been able to extrapolate something useful for the generalized problem.