Prime and Nonprime Totatives: A Sharp Construction and Exact Thresholds

For $n\ge 2$, let $A(n)=\pi (n)-\omega (n)$ and $B(n)=\phi (n)-\pi (n)+\omega (n)$ denote the numbers of prime and nonprime totatives of $n$, respectively, with $1$ included in $B(n)$. We construct an explicit injectively parametrized family ${\mathcal{C}}_n$ of composite totatives by completing suitably restricted squarefree products of prime totatives with a larger prime totative that is the unique largest prime factor, making the parametrization recoverable. We prove $\mid {\mathcal{C}}_n\mid \mathrm{/}A(n)\ge (e^{-\gamma }-o(1))\log A(n)\mathrm{/}\log \log A(n)$, realizing the sharp classical leading constant $e^{-\gamma }$. For the same family, with the same defining parameters, we obtain an effective refinement with error of order $(\log \log A(n){)}^{-1\mathrm{/}4}$ and absolute effectively computable constants. Classical minimal-order results and the prime number theorem show that $e^{-\gamma }$ is optimal: no family contained in the nonprime totatives can satisfy a uniform lower bound at this scale with a larger leading constant. We also determine exact eventual thresholds. Let $N_k$ and $M_k$ be the least integers such that $\phi (n)>k\pi (n)$ for every $n\ge N_k$ and $B(n)>kA(n)$ for every $n\ge M_k$, respectively. We prove $M_k\le N_{k+1}$ for every $k\ge 1$, determine $N_1,\dots ,N_6$ and $M_1,\dots ,M_5$ exactly, and obtain $M_k=N_{k+1}$ for $1\le k\le 5$.