Consider the sequence $(a_k)_{k\geq 1}$ of positive rational numbers defined by $a_1 = \frac{2020}{2021}$ and for $k \geq 1,$ if $a_k = \frac{m}{n}$ for relatively prime positive integers $m$ and $n,$ then$$a_{k+1} = \frac{m+18}{n+19}.$$Determine the sum of all positive integers $j$ such that the rational number $a_j$ can be written in the form $\frac{t}{t+1}$ for some positive integer $t.$
