How is the status of a bigger number known while the smaller not?

Question

In this wolfram mathworld article in the table below the status of $2059$ is known before the some smaller numbers, or they choose to aim the computation for that specific number?

Answer 1

Maybe this can be compared with the sitauation that it is trivial to find the complete prime factorization of $2^{1000}$, trivial to exhibit at least a partial factorization of $2^{1000}-9$, but things are much more difficult for the smaller number $2^{1000}-97$ (I guess).

Sometimes you are just lucky when finding a prime factor, and the results suggests that looking for factors $(2r+1)\cdot 2^{k+2}+1$ is a promising method. It may be the case that for the unknown cases, $r$ is unfeasibly large (cf. the much smaller case $k=5$, which has a much larger $r$), or no prime factor has this specific form, or the numbers are in fact prime. Note that even performing Lucas-Lehmer and similar tests is difficult with these numbers, though something like that seems to have been done for $k=4$ - but that is a very small number!