58th Ukrainian National Mathematical Olympiad

Let $a_1=p$ is prime. Then it is clear that $$\begin{array}{ccc}{a}_2⊢p,& a_3⊢a_2{a}_1⊢p^2,& a_4⊢a_3{a}_2{a}_1⊢p^4,\\ a_5⊢a_4{a}_3{a}_2{a}_1⊢p^8,& a_6⊢a_5{a}_4{a}_3{a}_2{a}_1⊢p^{16}.& \end{array}$$ Assume the length of the sequence is greater than 6, thus $a_7⊢a_6{a}_5{a}_4{a}_3{a}_2{a}_1⊢p^{32}$, there is a prime number that divides $a_7$ and it is included with degree not less than 32. By conditions on $a_n$ that is not possible. Therefore, the maximum length is $n=6$. The only prime number that is included with degree 16 in $a_6=2^2\cdot 3^3\cdot 5^5\cdot 7^7\cdot 11^{11}\cdot 13^{13}\cdot 17^{17}$ is $p=17$. This is the maximum value of $a_1$.

It suffices to show now, that such sequence exists. Take $$a_1=17,a_2=17,a_3=17^2,a_4=17^4,a_5=17^8,a_6=2^2\cdot 3^3\cdot 5^5\cdot 7^7\cdot 11^{11}\cdot 13^{13}\cdot 17^{17}.$$