Infinite families of noncototients

For any positive integer ${\displaystyle n}$ let ϕ(n) be the Euler function of n. A positive integer ${\displaystyle n}$ is called a noncototient if the equation x-ϕ(x)=n has no solution x. In this note, we give a sufficient condition on a positive integer k such that the geometrical progression ${\displaystyle {\left(2^{m}k\right)}_{m\ge 1}}$ consists entirely of noncototients. We then use computations to detect seven such positive integers k.