A question about some polynomial

Question

Let $p_n$ be the $n$-th prime. Then $$ p_n = 1+\prod_{i=1}^r{p_{k_i}^{a_i}} $$ with $k_i < n$, $1 \le i \le r$ since this is just the prime factorization of $p_n-1$. Consider the polynomial: $$f_n = 1+\prod_{i=1}^r{X_{k_i}^{a_i}} $$ Computing this polynomial for small values of $n$ it seems that this polynomial is irreducible over $\mathbb{Q}$. Is there any reason for this, or might this polynomial be also reducible?