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Integer sequence and prime divisors

Source: Turkey TST 2019 Day 1 P2

March 26, 2019
number theorySequenceprime numbers

Problem Statement

(an)n=1∞(a_{n})_{n=1}^{\infty} is an integer sequence, a1=1a_{1}=1, a2=2a_{2}=2 and for n≥1n\geq{1}, an+2=an+12+(n+2)an+1−an2−nana_{n+2}=a_{n+1}^{2}+(n+2)a_{n+1}-a_{n}^{2}-na_{n}. a)a) Prove that the set of primes that divides at least one term of the sequence can not be finite. b)b) Find 3 different prime numbers that do not divide any terms of this sequence.
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