*Gleason's polynomials* are the sequence of monic integer polynomials defined recursively by
$$
\prod_{d \mid n} G_d(c) = (((c^2+c)^2+c)^2+\cdots+c)^2+c \quad \quad \quad [\textrm{$n$ iterates}],
$$
for $n=1,2, \ldots$. Thus they start out like:
$$
G_1 = c, \quad G_2 = c+1, \quad G_3  = c^3 + 2c^2 + c + 1, \quad  G_4(c) =  c^6 +  3c^5 +  3c^4  + 3c^3 + 2c^2 + 1, \, \ldots
$$
They give the period-$n$ centers for the hyperbolic components of the Mandelbrot set in complex dynamics. In many ways they resemble the cyclotomic polynomials, which would result if we had $c^n-1$ on the right-hand side of the recursive definition; or the dynatomic polynomials in the dynamical plane (this is their version in the parameter plane parametrizing the quadratic iterations $z^2+c$). 

For example, $\mathrm{Res}(G_n,G_m) = \pm 1$ for any pair $n \neq m$, just as for the cyclotomic polynomials. This is proved for instance in Corollary 4.8 of [this paper by Hutz and Towsley][1]. (Courtesy of Matt Baker for this reference. *An aside:* What are the exact signs?) 

My question is about the discriminants of these polynomials:  

*What are the lower and upper growth rates of $\delta_n := \log{|\mathrm{Disc}(G_n)|}$? Does $\frac{1}{n}\sum_{d \mid n} \delta_d \asymp \log{n}$?* 

(The latter $\sim \log{n}$ for the cyclotomics, and the upper and lower growth rates $\sim \phi(n)\log{n}$ for the individual cyclotomic discriminants are $n\log{n}$ and $e^{-\gamma}n\frac{\log{n}}{\log{\log{n}}}$, by Mertens's theorem.)

I was wondering in what ways would the Gleason discriminants behave similarly to the cyclotomic discriminants (size-wise?), and in what ways they are markedly different. One marked difference is that the prime factors of the $\mathrm{Disc}(G_n)$ are quite unpredictable; a casual look at the first few prime factorizations of the discriminants of $G_3, G_4, G_5$ and $G_6$ reveals 
$$
23 \times 2551, \quad 13 \times 24554691821639909, \quad 13^2 \times 949818439 \times 6488190752068386528993226361, \quad 8291 × 9137 × 420221 × 189946 395389 × 4813 162343 551332 730513 × 2 837919 018511 214750 008829 × 1 858730 157152 877176 856713 108209 153714 699601
$$

The one thing that is easy and very useful to see is that these discriminants are odd: this is how Gleason established that the complex roots of the polynomials are all distinct (no multiplicities), which is non-obvious from the definition. 

  [1]: https://arxiv.org/pdf/1309.4048.pdf