On the upper and lower bound orders of almost prime triples

Abstract: 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.