  We obtain uniform estimates for $N_k(x,y)$, the number of positive integers
$n$ up to $x$ for which $\omega_y(n)=k$, where $\omega_y(n)$ is the number of
distinct prime factors of $n$ which are $<y$. The motivation for this problem
is an observation due to the first author in 2015 that for certain ranges of
$y$, the asymptotic behavior of $N_k(x,y)$ is different from the classical
situation concerning $N_k(x,x)$ studied by Sathe and Selberg. We demonstrate
this variation of the classical theme; to estimate $N_k(x,y)$ we study the sum
$S_z(x,y)=\sum_{n\le x}z^{\omega_y(n)}$ for $Re(z)>0$ by the Buchstab-de Bruijn
method. We also utilize a certain recent result of Tenenbaum to complete our
asymptotic analysis.
