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.
Categories: math.NT