Let $N = \{n \in \mathbb{Z}^+ \mid n = p_1^{\alpha_1}p_2^{\alpha_2}\dots p_k^{\alpha_k}\}$, where $p_i$ is the $i$th prime number and $\alpha_i \in \mathbb{N}_0$. Suppose that we are given some arbitrary $n \in N$. Does there exist a method for minimising $\delta = |n - n'| > 0$, where $n'$ also lies in $N$? (In particular, the condition that $\delta > 0$ implies $n \neq n'$).
I've looked around and can't find any problems similar to this one. Any insight into how this problem may be tackled is appreciated. I'm mainly interested in behaviour for large $n$, for some fixed $k$.
Edited for inclusion of example:
Suppose I consider the set $P= \{2,3,5,7\}$. If I considered $n = 49$, then $\delta = 1$ since $50 = 2\cdot 5^2$ (or alternatively, because $48 = 2^4\cdot 3$). However, for $n = 98$, we have $\delta = 2$ since $99 = 3^2\cdot 11$ and $11 \notin P$, and $97$ is itself prime and not contained in $P$. For $n = 196$, it turns out that $\delta = 4$, and for $n = 392$, we have $\delta = 8$. I suspect that for large values of $n$, the value of $\delta$ should also be large.