Consider integers of the form $$n=p_1^{k_1}\cdots p_l^{k_l},$$ with $p_{i}$ primes which equal $1\mod 3$. With help of Mathematica, it seems that there exist quadratic families of subsets of values of $n$. For example, a family of values of $n$ is given by $$n_m=1+12m^2,$$ which in particular contains $13$, $49$, etc, which are of the form stated above. I have been able to find a number of other families of quadratic curves $n^{(i)}_m$.
My question is the following: do all $n$ of the above stated form lie on some quadratic curve? And is there a general formula which combines all such curves in a single expression?