On the denominators of generalized harmonic numbers. II

For three positive integers $a$, $b$ and $n$, let $H_{a,b}(n)$ be the sum of the reciprocals of the first $n$ terms of arithmetic progression $\{ak+b:k=0,1,\dots\}$, and let $v_{a,b}(n)$ be the denominator of $H_{a,b}(n)$. In this paper, we prove that the set of positive integers $n$ satisfying $\nu_p(v_{a,b}(n))=\nu_p([b,b+a,\dots,b+(n-1)a])$ has positive logarithmic density, where $[b,b+a,\dots,b+(n-1)a]$ denotes the least common multiple of $b,b+a,\dots,b+(n-1)a$.