For each positive integer $n$, let $f(n)$ be the number of positive integers pairs $(a,b)$ such that $a+i$ divides $b+i$ for $i = 1,2,...,n$. What is the growth rate of $f(n)$?
For each $n$ there are infinitely many pairs $(a,b)$ satisfying the requirement. We can simply take $a=1$. This requires $n$ lines of $$b+1 \equiv 0 \pmod 2 \\b+2 \equiv 0 \pmod 3\\b+3 \equiv 0 \pmod 4$$ and so on. The Chinese remainder theorem guarantees an infinite number of solutions.