What is the solution to \min_k k \frac{b^k/n}{\lfloor b^k/n \rfloor}

Question

(I've posted this question at Math.SE but got no answer, so I hope I can get a solution here.)

This problem looks familiar, but I don't remember its solution:

$$ \min_k \ \ \frac{b^k/n}{\lfloor b^k/n \rfloor}k $$

subject to

$$ b^k \ge n \\ b,n,k \in \mathbb{N} $$

Does it have a name? What's the solution? Thanks!