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Sequence formed by product of prime factors

Source: Simon Marais MC 2019 A1

October 14, 2019
number theory

Problem Statement

Consider the sequence of positive integers defined by s1,s2,s3,…s_1,s_2,s_3, \dotsc of positive integers defined by
[*]s1=2s_1=2, and[/*] [*]for each positive integer nn, sn+1s_{n+1} is equal to sns_n plus the product of prime factors of sns_n.[/*]
The first terms of the sequence are 2,4,6,12,18,242,4,6,12,18,24. Prove that the product of the 20192019 smallest primes is a term of the sequence.
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