2021 AIME I Problem 10

Consider the sequence \left(a_{k}\right)_{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} \text {. }

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.