  Let $S=K[x_1,\ldots,x_n]$ be the polynomial ring in $n$ variables over a
field $K$. In this paper, we compute the socle of $\cb$-bounded strongly stable
ideals and determine that the saturation number of strongly stable ideals and
of equigenerated $\cb$-bounded strongly stable ideals. We also provide explicit
formulas for the saturation number $\sat(I)$ of Veronese type ideals $I$. Using
this formula, we show that $\sat(I^k)$ is quasi-linear from the beginning and
we determine the quasi-linear function explicitly.
