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Recursion And Primes

Source: Bundeswettbewerb Mathematik 2017, Round 1 - #4

August 7, 2017
number theorynumber theory unsolvedSequencerecursionprime numbersInteger

Problem Statement

The sequence a0,a1,a2,…a_0,a_1,a_2,\dots is recursively defined by a_0 = 1   \text{and}   a_n = a_{n-1} \cdot \left(4-\frac{2}{n} \right)   \text{for } n \geq 1. Prove for each integer n≥1n \geq 1:
(a) The number ana_n is a positive integer. (b) Each prime pp with n<p≤2nn < p \leq 2n is a divisor of ana_n. (c) If nn is a prime, then an−2a_n-2 is divisible by nn.
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