Level Totients for Integer Mosaics

We study a level analog of Euler’s totient function for integer mosaics. Let $P_i(n)$ be the set of primes appearing in the first $i$ levels of the mosaic of $n$, and let ${\phi }_i(n)$ count the integers $m\le n$ for which $P_i(m)\cap P_i(n)=\mathrm{\varnothing }$. We prove a Möbius divisor-sum formula for ${\phi }_i(n)$ and reduce it to a sum over a set ${\mathcal{V}}_{i,S}$ of powerful integers. If $S\mathrm{\ne }\mathrm{\varnothing }$, $i\ge 2$, and $q=\min S$, then $$\mathrm{\mid }{\mathcal{V}}_{i,S}\cap [1,x]\mathrm{\mid }\sim C_{i,S}{x}^{1\mathrm{/}q},$$ with $C_{i,S}$ an explicit positive Euler-product constant. For fixed $S$, the density ${\delta }_i(S)$ of integers whose first $i$ levels avoid $S$ exists and has an Euler product; for nonempty $S$, $i\ge 2$, and $q=\min S$, the number of such integers up to $N$ is ${\delta }_i(S)N+O_{i,S}(N^{1\mathrm{/}q})$. Taking $S=P_i(n)$ gives $$\frac{{\phi }_i(n)}{n}={\delta }_i(P_i(n))+O_{\epsilon }(n^{-1\mathrm{/}2+\epsilon })$$ uniformly in $n$.