FKMO 2017 Problem 4

For a positive integer $n \ge 2$ , define a sequence $a_1, a_2, \cdots ,a_n$ as the following. $$a_1 = \frac{n(2n-1)(2n+1)}{3}$$ $$a_k = \frac{(n+k-1)(n-k+1)}{2(k-1)(2k+1)}a_{k-1}, \text{ } (k=2,3, \cdots n)$$ (a) Show that $a_1, a_2, \cdots a_n$ are all integers.

(b) Prove that there are exactly one number out of $a_1, a_2, \cdots a_n$ which is not a multiple of $2n-1$ and exactly one number out of $a_1, a_2, \cdots a_n$ which is not a multiple of $2n+1$ if and only if $2n-1$ and $2n+1$ are all primes.