SIMON-MARAIS 2019 A1

Consider the sequence of positive integers defined by $s_1,s_2,s_3, \dotsc$ of positive integers defined by

• $s_1=2$ , and[/*]
• for each positive integer $n$ , $s_{n+1}$ is equal to $s_n$ plus the product of prime factors of $s_n$ .[/*]

The first terms of the sequence are $2,4,6,12,18,24$ . Prove that the product of the $2019$ smallest primes is a term of the sequence.