Let $H$ be the multiplicative monoid of the (usual) semiring of polynomials in one variable $x$ with coefficients in $\mathbf N$. Given $\alpha, k \in \mathbf N$, denote by $\mathcal A_k(\alpha)$ the set of all atoms of $H$ of the form $c_0 + \cdots + c_k x^k$ such that $c_0 = c_k = 1$ and $c_i \le \alpha$ for $i \in [\![1,k-1]\!]$. (A polynomial $f \in H$ is an atom if $f \ne 1$ and there don't exist $g, h \in H \setminus \{1\}$ for which $f = gh$.)
It seems plausible to me that someone else has already investigated the asymptotic behavior of $a_k(\alpha) := |\mathcal A_k(\alpha)|$ as a function of $k$ (in the limit as $k \to \infty$). Any pointer? Should it be helpful, I'm mostly interested in the regime $\alpha \sim k$ as $k \to \infty$.
Of course, $a_k(\alpha) \le a_k(\alpha+1)$ for all $\alpha, k \in \mathbf N$. As for the analysis of very small or trivial cases, we have $a_0(\alpha) = 0$ and $a_k(0) = a_1(\alpha) = 1$ for $k \ge 1$ and every $\alpha \in \mathbf N$; moreover, it is easily seen that $a_2(1) = a_2(2) = 2$ and $a_2(\alpha) = \alpha - 1$ for $\alpha \ge 3$.
