Say that the $n$-th prime $p_n$ is isolated to degree $k$ (my notation) if the prime gap to either side is larger than $p_n$ to the $k$-th power: \begin{eqnarray*} p_n - p_{n-1} & > & p_n^k \;,\\ p_{n+1} - p_n & > & p_n^k \;. \end{eqnarray*} Examples.

• For $k=0.4$, $p_{62}=293$ is isolated because $(p_{n-1},p_n,p_{n+1}) = (283,293,307)$, and the gaps of $10$ and $14$ both exceed $293^{0.4} \approx 9.7$.

• For $k=0.3$, $p_{34871}=413353$ is isolated because $(p_{n-1},p_n,p_{n+1}) = (413299,413353,413411)$, and the gaps of $54$ and $58$ both exceed $413353^{0.3} \approx 48.4$.

Q. For which $k$ (if any) are there an infinite number of isolated primes of degree $k$?

Among the first $10$ million primes, $293$ is the largest isolated prime of degree $0.4$, and $413353$ is the largest isolated prime of degree $0.3$. And there are no isolated primes of degree $\frac{1}{2}$ among the first $10$ million primes. (The $10$-th million prime is $179424673$.)