On the upper and lower bound orders of almost prime triples

A Hardy-Littlewood triple is a 3-tuple of integers with the form $(n,n+2,n+6)$. In this paper, we study Hardy-Littlewood triples of the form $(p,P_a,P_b)$ and improve the upper and lower bound orders of it, where $p$ is a prime and $P_r$ has at most $r$ prime factors. Our new results generalize and improve the previous results.