Question 145
What is the maximum length of a sequence of positive integers $a_1, a_2, \dots, a_n$, if following conditions hold:
* $a_1 \gt 1$ is a prime number;
* for any $i: 2 \le i \le n$, $a_i$ is divisible by $a_1 a_2 \dots a_{i-1}$;
* $a_n = 2^2 \cdot 3^3 \cdot 5^5 \cdot 7^7 \cdot 11^{11} \cdot 13^{13} \cdot 17^{17}$.
* $a_1 \gt 1$ is a prime number;
* for any $i: 2 \le i \le n$, $a_i$ is divisible by $a_1 a_2 \dots a_{i-1}$;
* $a_n = 2^2 \cdot 3^3 \cdot 5^5 \cdot 7^7 \cdot 11^{11} \cdot 13^{13} \cdot 17^{17}$.