Is there an analogue to Bertrand's Postulate for primes for perfect powers?
Bertrand's Postulate: $\forall x \gt 1 \in \mathbb{Z}, \exists p \in \mathbb{P}$ such that $x < p < 2x$.
For perfect powers, could we say something like $\exists k,m,\lambda \in \mathbb{Z}, m \gt 2$ such that $\forall x \in \mathbb{Z}, x \gt 2$, $x \lt k^m \le \lambda x$? Obviously if $\lambda \ge x^2$, we have $(k,m) = (x, 3)$ as a satisficatory perfect power in the range.
This question came up while thinking about this other MSE question I had posted and received an answer. I am interested in knowing how close to $x$ are we likely to find a perfect power $k^m \gt x$ with an exponent $m\gt2$?