Who's next to me in the queue?
Perl 6, 79 76 71 bytes



{($!=sort {[R,] -$_,{first $_%%*,$_^..1}...1},^2020)[+(@$!...$_)X-2,0]}


Try it online!

Sorts numbers 1 to 2019 lexicographically by the reversed, negated sequence of largest divisors, then finds the neighbors. Example mappings:

1760 =&gt; (-1 -11 -55 -110 -220 -440 -880 -1760)
  33 =&gt; (-1 -11 -33)
  99 =&gt; (-1 -11 -33 -99)


In other words, the negated cumulative product of prime factors in descending order.