Given an $\alpha\in(0,1)$ and $n\in\Bbb N$ what are some known deterministic algorithms to sample $O(n^\alpha)$ (not just get one) square free integers of $n$ bits? Is it $O(n^{\alpha})$ complexity? By sample I mean can we get certain number so that atleast some fraction of them are some distance from each other.

Given an $\alpha\in(0,1)$ and $n\in\Bbb N$ what are some known probabilistic algorithms to sample $O(n^\alpha)$ square free integers of $n$ bits that need not be primes? Is it $O(n^{\alpha})$ complexity?

Is it known if testing a given interval $[a,b]$ contains a prime is in $BPP$ (assume each $a$ and $b$ have $n$ bits)?

I want the certificate for square free numbers that are generated to be in poly time. Note we may not have a square free certificate without factoring but that does not preclude certification for outputs of particular algorithms to be in poly time (these numbers could be output in special ways or specially constructed).