Problem: Given $n = ab > 0$ with $a,b \in \mathbb{Z}$ unknown, what are some good choices for $a_0, d$ such that $a,b$ are both in the AP $a_0 + kd$? We are not making any assumptions about $a,b$ being prime. So, we need to consider all divisors.
Obvious choices are
$$ (a_0,d) = \begin{cases} (1,2), & \text{if $n$ is odd}, \\ (0,1), & \text{$\forall n$}, \\ (1,1), & \text{$\forall n$}. \\ \end{cases} $$
The second and third formulations are the same AP and are equivalent.
Therefore, a good choice would be the pair $(a_0, d)$ such that $d > 2$. Are there other choices that depend on $n$, perhaps based on modular conditions etc.,?
In general, $a$ and $b$ are in the arithmetic progression $c+kd$ if and only if $d\mid a-b$ and $c\equiv a \pmod d$. This gives all the solutions.