Let $n$ be a natural number, $\zeta_{n}$ be a primitive $n^{th}$ root of unity and $\Phi_{n}(x)$ be the $n^{th}$ cyclotomic polynomial. Let $\Psi_{n}(x)$ be the $n^{th}$ real cyclotomic polynomial (which is defined to be the minimal polynomial of $\zeta_{n} + \zeta_{n}^{-1}$).
We know that $ x^{n} - 1 = \prod_{d \mid n} \Phi_{d}(x) $. I vaguely recall reading a few years ago that there is such polynomial $G_{n}(x)$ such that $G_{n}(x) = \prod_{d \mid n, d> 1} \Psi_{d}(x)$. In fact, if I am not fabricating my memory, this $G_{n}(x)$ was denoted by $\chi_{n}(x)$.
Can anyone help me by pointing me in right direction so that I can recover such a relation, if one exists?