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it's probably most like the collatz conjecture, a question about the behavior of a system over time. let $n= $ at $t_ $. at each $t_i$, if either $n+i$ or $n-i$ is prime, but not both, then set $n=n+i$ or $n-i$, respectively. also, $n$ cannot visit the same prime twice, i.e. can never repeat a value.
note it won't change to $ $ or $ $ at $t_ $, so the next stop is $n= $ at $t_{ }$. a graph of $n$ over the first couple million $t$. note that at this point, only about half of the primes $< $ have been hit. the main question is... as $t ightarrow \infty$, will $n$ eventually take every prime value or not...
https://math.stackexchange.com/questions/3545124/collatz-esque-dynamical-problem-about-prime-distribution