A familiar scent reaches your nose, but you can't recall its origin. Maybe you have smelt it once or twice in your life; it's strong and not too pleasant. To open the large boulder gate before you, you need something: not a measly chant, but a computation.

Consider a polynomial $p(x)$ of degree $n>1$ with real coefficients, written in the form $$ p(x) = a_0 + a_1x + a_2x^2 + \ldots + a_nx^n. $$ Define the corresponding $p^*(x)$ as follows. $$ p^*(x) = a_n + a_{n-1}x + a_{n-2}x^2 + \ldots + a_0x^n. $$ Now, say that a natural number $k$ has the prime factorization $$ k=p_1^{r_1}p_2^{r_2}\ldots p_j^{r_j}, $$ where the $p_i$ are distinct primes and $r_i$ are their maximum powers in $k$. Finally, define $$ f(x,k) = k(p_1 - x)(p_2 - x)\ldots (p_j - x). $$ Note that this is clearly a polynomial of degree $j$; say $j > 1$.

Find the square of the following quantity. $$ \frac{f^*(1,k)}{p_1p_2 \cdots p_j \cdot \phi (k)}-18. $$ Here, $\phi$ is the Euler totient function.