Let's introduce Sergei (for SElf-Referential Gaps Extensible to Infinity, and as a wink to a mathematician friend of mine of Russian descent whose given name is Serge and quite interested in number theory) numbers as even integers $ n $ such that $ n $ can be written in at least $ n $ ways as the difference between consecutive odd primes. For example $ 2=13-11=7-5$ is a Sergei number. 


Are there provably infinitely many Sergei numbers ? What about the best reachable lower bound of their density among even numbers ? 



Sergei numbers : even integers n being a prime gap at least n times
In fact, even stronger fact is true: every Polignac number is a Sergei number and therefore there is a constant $C$ such that every interval of the form $[M,M+C]$ contains at least one Sergei number.
The corresponding result for Polignac numbers is proved in [this](https://arxiv.org/abs/1305.6289) article, Theorem 2.


