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