  Let $P^+(n)$ denote the largest prime factor of the integer $n$ and
$P_y^+(n)$ denote the largest prime factor $p$ of $n$ which satisfies
$p\leqslant y$. In this paper, firstly we show that the triple consecutive
integers with the two patterns $P^+(n-1)>P^+(n)<P^+(n+1)$ and
$P^+(n-1)<P^+(n)>P^+(n+1)$ have a positive proportion respectively. More
generally, with the same methods we can prove that for any $J\in \mathbb{Z},
J\geqslant3$, the $J-$tuple consecutive integers with the two patterns
$P^+(n+j_0)= \min\limits_{0\leqslant j\leqslant J-1}P^+(n+j)$ and $P^+(n+j_0)=
\max\limits_{0\leqslant j\leqslant J-1}P^+(n+j)$ also have a positive
proportion respectively. Secondly for $y=x^{\theta}$ with $0<\theta\leqslant 1$
we show that there exists a positive proportion of integers $n$ such that
$P_y^+(n)<P_y^+(n+1)$. Specially, we can prove that the proportion of integers
$n$ such that $P^+(n)<P^+(n+1)$ is larger than 0.1356, which improves the
previous result "0.1063" of the author.
