Constructing normal numbers using residues of selective prime factors of integers

Abstract. Given an integer \(N\ge 1\), for each integer \(n\in J_N:= [e^N,e^{N+1})\), let \(q_N(n)\) be the smallest prime factor of \(n\) which is larger than \(N\); if no such prime factor exists, set \(q_N(n)=1\). Fix an integer \(Q\ge 3\) and consider the function \(f(n)=f_Q(n)\) defined by \(f(n)=\ell\) if \(n \equiv \ell \pmod Q\) with \((\ell,Q)=1\) and by \(f(n)=\Lambda\) otherwise, where \(\Lambda\) stands for the empty word. Then consider the sequence \((\kappa(n))_{n\ge 1}=(\kappa_Q(n))_{n\ge 1}\) defined by \(\kappa(n)= f(q_N(n))\) if \(n\in J_N\) with \(q_N(n)>1\) and by \(\kappa(n)= \Lambda\) if \(n\in J_N\) with \(q_N(n)=1\). Then, for each integer \(N\ge 1\), consider the concanetation of the numbers \(\kappa(1), \kappa(2), \ldots\), that is define \(\theta_N:=\mbox{Concat}(\kappa(n):n\in J_N)\). Then, set \(\alpha_Q:= \mbox{Concat}(\theta_N:N=1,2,3,\ldots)\). Finally, let \(B_Q=\{\ell_1,\ell_2,\ldots,\ell_{\varphi(Q)}\}\) be the set of reduced residues modulo \(Q\), where \(\varphi\) stands for the Euler function. We show that \(\alpha_Q\) is a normal sequence over \(B_Q\).