Useful Uspensky

$2020, \lfloor \alpha \rfloor, \lfloor \beta \rfloor, 4040, \lfloor 2\alpha \rfloor, \lfloor 2\beta\rfloor, 6060, \lfloor 3\alpha \rfloor, \lfloor 3\beta\rfloor, \dots$

Do there exist two positive real numbers $\alpha$ and $\beta$ , such that each positive integer appears exactly once in the sequence above?

This problem is from Korea 2020 Math Contest.