  For two relatively prime square-free positive integers $a$ and $b$, we study
integers of the form $a p+b P_{2}$ and give a new lower bound for it, where $a
p$ and $b P_{2}$ are both square-free, $p$ denotes a prime, and $P_{2}$ has at
most two prime factors. We also consider some special cases where $p$ is small,
$p$ and $P_2$ are within short intervals, $p$ and $P_2$ are within arithmetical
progressions and a Goldbach-type upper bound result. Our new results generalize
and improve the previous results.
