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 $\log p_n$ to the $k$-th power:
\begin{eqnarray*}
p_n - p_{n-1} & > & (\log p_n)^k \;,\\
p_{n+1} - p_n & > & (\log p_n)^k \;,
\end{eqnarray*}
where $\log$ is the natural log.

*Examples*.

For $k=1.5$, $p_{4059}=38501$ is isolated because $$(p_{n-1},p_n,p_{n+1}) = (38461,38501,38543)\;,$$ and the gaps of $40$ and $42$ both exceed $(\log 38501)^{3/2} \approx 10.6^{1.5} \approx 34.3$.

For $k=1.6$, $p_{722697}=10938023$ is isolated because $$(p_{n-1},p_n,p_{n+1}) = (10937921,10938023,10938119)\;,$$ and the gaps of $102$ and $96$ both exceed $(\log 10938023)^{1.6} \approx 86.2$.

For $k=1.7$, I find no isolated primes in the first $10$-million primes. (The $10$-th million prime is $179424673$.)

Among the first $10$-million primes, about $13$% are isolated to degree $k = 1$, and $73$% are isolated to degree $k=\frac{1}{2}$.

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

1more comment