  In the present paper, we study the Dao numbers
$\mathfrak{d}_1(I),\mathfrak{d}_2(I)$ and $\mathfrak{d}_3(I)$ of an ideal $I$
of a Noetherian local ring $(R,\mathfrak{m},K)$ or a standard graded Noetherian
$K$-algebra. They are defined as the smallest $\ell\ge0$ such that
$I\mathfrak{m}^k$ is $\mathfrak{m}$-full, full, weakly $\mathfrak{m}$-full,
respectively, for all $k\ge\ell$. We provide general bounds for the Dao numbers
in terms of the Castelnuovo-Mumford regularity of certain modules over the Rees
algebra $\mathcal{R}(\mathfrak{m})$. If $R$ is a Koszul algebra, we prove that
the Dao numbers are less or equal to
$\text{reg}_{\text{gr}_\mathfrak{m}(R)}\text{gr}_\mathfrak{m}(I)$, where
$\text{gr}_\mathfrak{m}(I)$ is the associated graded module of $I$. Finally,
for monomial ideals, we combinatorially bound the Dao numbers in terms of
asymptotic linear quotients and bounding multidegrees.
