关于乘积整除性的特定递增数列存在性问题问询

Hey everyone, here's an interesting number theory question a classmate of mine posed:

Whether for any positive integer $n$, there exists a sequence of strictly increasing positive integers $0<t_1<t_2<...<t_n$ such that for any pair of integers $0<a<b<\text{lcm}(t_1,t_2,...,t_n)$, the product $\prod_{i=1}^n(a+t_i)$ does not divide $\prod_{i=1}^n(b+t_i)$

I've tested this for small values of $n$ myself, and found that such a sequence exists for every $n \le 18$. For the case $n=18$, the sequence:
(1,2,3,4,5,6,8,10,12,15,20,24,30,48,60,80,120,240)
works perfectly as a solution.

备注：内容来源于stack exchange，提问作者Rogerhu