Question 177
The sequence $a_1, a_2, a_3, \dots$ of positive integers is defined in the following way: $a_1$ is given, and for each $n \ge 2$ the number $a_n$ is the smallest positive integer divisible by $n$, which is not less than $a_{n-1}$. (For example, if $a_5 = 115$, then $a_6 = 120, a_7 = 126, a_8 = 128$.)
Prove that if $a_1 \le 2019$, then each positive integer can occur in the sequence $a_{63}, a_{64}, a_{65}, \dots$ at most once.
Prove that if $a_1 \le 2019$, then each positive integer can occur in the sequence $a_{63}, a_{64}, a_{65}, \dots$ at most once.