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exists index i, such prime p|a_i, a_{n+1}=a_n^2 + n a_n .

Source: 2022 Saudi Arabia IMO TST 1.1

November 1, 2022
number theorydividesnumber theory with sequencesrecurrence relation

Problem Statement

Let (an)(a_n) be the integer sequence which is defined by a1=1a_1= 1 and an+1=an2+n⋅an  ,  ∀n≥1. a_{n+1}=a_n^2 + n \cdot a_n \,\, , \,\, \forall n \ge 1. Let SS be the set of all primes pp such that there exists an index ii such that p∣aip|a_i. Prove that the set SS is an infinite set and it is not equal to the set of all primes.
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