Is it computable if a particular number follows the Collatz conjecture?
In short: we don't know :-)

There is a ((very) little) chance that the Collatz sequence is Turing complete (with probably some caveats like in the case of Two-counter Machines); i.e. there is an algorithm which for every turing machine $M$ outputs an $x$ such that:

$Collatz(x)$ is on a divergent trajectory if and only if $M$ doesn't halt on empty tape

in other words "deciding if a particular Collatz trajectory is divergent" could be undecidable.