DE-BUNDESWETTBEWERB 2020 Problem 4

Define a sequence $(a_n)$ recursively by $a_1=0, a_2=2, a_3=3$ and $a_n=\max_{0<d<n} a_d \cdot a_{n-d}$ for $n \ge 4$ . Determine the prime factorization of $a_{19702020}$ .