  Let $${\mathbb P}^c=(\lfloor p^c\rfloor)_{p\in{\mathbb P}} \qquad (c>1,\
c\not\in {\mathbb N}), $$ where ${\mathbb P}$ is the set of prime numbers, and
$\lfloor\cdot\rfloor$ is the floor function. We show that for every such $c$
there are infinitely many members of ${\mathbb P}^c$ having at most $R(c)$
prime factors, giving explicit estimates for $R(c)$ when $c$ is near one and
also when $c$ is large.
