  We consider the equation $[p_{1}^{c}] + [p_{2}^{c}] + [p_{3}^{c}] = N$, where
$N$ is a sufficiently large integer, and prove that if $1 < c < \frac{17}{16}$,
then it has a solution in prime numbers $p_{1}$, $p_{2}$, $p_{3}$ such that
each of the numbers $p_{1} + 2$, $p_{2} + 2$, $p_{3} + 2$ has at most $\left [
\frac{95}{17 - 16c} \right ]$ prime factors, counted with the multiplicity.
