Let $P(n)$ denote the product of the prime divisors of $n$, e.g., $P(100)=2\times 5=10$. Define $$A=\{\frac{a}{P(ab(a-b))} \mid a,b\in\mathbb{Z}^+, a>b\}.$$ Is $A$ bounded or not?
To make the value $\frac{a}{P(ab(a-b))}$ high, we can choose $a$ to be a perfect power, such as $a=2^n$, and $b$ can be some constant. This would suffice if $a-b$ consists of only small prime factors for some large $n$.
Take $a=2\cdot 3^n,$ $b=3^n.$ Then the term becomes $3^{n-1}.$
[I assume from your example $P$ only is product of the distinct primes dividing $n,$ and note with these choices $P(ab(a-b))=2\cdot 3.$]