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$a_{n+1}=(a_1+a_2+\cdots+a_n)^2-1$

Source: Thailand Online MO 2021 P3

April 5, 2021
Sequencenumber theory

Problem Statement

Let a1,a2,⋯a_1,a_2,\cdots be an infinity sequence of positive integers such that a1=2021a_1=2021 and an+1=(a1+a2+⋯+an)2−1a_{n+1}=(a_1+a_2+\cdots+a_n)^2-1 for all positive integers nn. Prove that for any integer n≥2n\ge 2, ana_n is the product of at least 2n2n (not necessarily distinct) primes.
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