In 2000, Baker, Harman and Pintz proved that there is always a prime in
the interval $(n-n^{0.525}, n)$. There are also conditional results
implying smaller intervals. Nevertheless, I could not find any
information about prime power gaps. So, what I'm asking is:
What is the asymptotically largest function $f(n)$ s.t. there is always a prime power in the interval $(f(n), n)$?
For example, Bertrand’s postulate is almost trivial in this case, since there is always a power
of $2$ in the interval $(n, 2n]$. On the other hand, the distribution of prime powers with
exponent $e>1$ is much smaller than the distribution of primes, so adding them might not change
the answer.
Best Answer
The paper shows that for $x \gt x_0$ there are primes in the interval $[x-x^{0.525},x]$ where the value of $x_0$ could be found effectively "with enough effort." The result fails for $x=126$ but I can't immediately rule out that $x_0=127$ suffices. The paper seems to say that for large enough $n$ the number of primes in $[n,n+n^{0.525}]$ eventually exceeds $\frac{9n^{0.525}}{100\log{n}}.$
Call an integer power proper if it is at least a square and respectable if it is at least a cube, To expand on Gerhard's observation: If we looked for intervals which contain either a prime or a proper integer power then we can say that there is always a square in $(n-2\sqrt{n},n)$ so this eventually improves on $x-x^{.525}.$
There is always a respectable power in $(n-3n^{2/3},n)$ but that is pretty near best possible, so I would guess that if we said "a prime or a respectable power" then we would not be able to get any improvement over just "prime". Likewise, there seems no reason to think that "prime or prime power" is essentially better than "prime or prime square" nor that "prime or prime square" is essentially better than "prime."