Define the following function function $$F_m(n) = -n + \prod_{p\mid n}\left(p^2-m\right)$$ for $n, m \in \mathbb{N}$. It is connected to the Dedekind psi function and Euler totient function as follows, Let $$\phi (n) = n\prod_{p\mid n}\left( 1 - \frac{1}{p}\right)$$ $$\psi (n) = n\prod_{p\mid n}\left( 1 + \frac{1}{p}\right)$$ Then $$\psi (n)\phi (n) = n^2\prod_{p\mid n}\left(1-\frac{1}{p^2}\right)$$ Which is equal to Jordan's totient function $J_2(n)$. We can rearrange to find $$J_2(n) = \left(\frac{n}{\text{rad}(n)}\right)^2\prod_{p\mid n}\left(p^2-1\right)$$ Which after rearranging leaves us with the special case that $$F_1(n) = J_2(n)\left(\frac{\text{rad}(n)}{n}\right)^2 - n$$
I'm interested in finding all numbers such that they are a root of $F_1(n)$. An obvious requirement for this is that $$n = \prod_{p\mid n}p^2-1$$ We can easily see by the right hand side $\prod_{p\mid n}p^2-1$ is independant of the total number of prime factors. If we consider $n = p^{\alpha}$ then we can see $$p^{\alpha} = p^2-1$$ which does not hold true. Hence we need $n: \omega(n) > 1$, where $$\omega(n) = \sum_{p\mid n}1$$ For example, for any number of the form $2^{\alpha} \cdot 3^{\beta}$ $\forall \alpha, \beta \in \mathbb{N}$, we get $$F_1(2^{\alpha} \cdot 3^{\beta})=(2^2-1)(3^2-1) = 24$$ And since $24 = 3 \cdot 2^3$ we get $$F_1(24) = 0$$
24 seems to be the smallest example. Following the method of increasing in value of prime divisors we can consider the next value to be of the form $2^{\alpha}\cdot 5^{\beta}$. $$F_1(2^{\alpha}\cdot 5^{\beta}) = 72$$ However $72 = 2^3 \cdot 3^2$ hence there is no $F_1(2^{\alpha}\cdot 5^{\beta}) = 72$. We can also see that there is no $F_1(3^{\alpha}\cdot 5^{\beta})$ as $5^2-1 = 3\cdot 2^3$. The same goes for $7^2-1 = 2(5^2-1) =...$. In fact, I don't know if this is proven or not, but it seems that all $$p^2-1 = 2^{\alpha} \cdot 3^{\beta} \cdot q \,\,\,\text{for some q} \in \mathbb{N}$$
Therefore, I conjecture that 24 is the only root of $F_1(n)$. My question is whether anyone has any ideas, approahes, things they would like to add... etc that you think might be useful, or if anyone has found a value for $n$ that would disprove the conjecture. I'm open to any suggestions too. Many thanks!