The number of large prime factors of integers and normal numbers

Abstract

In a series of papers, we constructed large families of normal numbers using the concatenation of the values of the largest prime factor P(n), as n runs through particular sequences of positive integers. A similar approach using the smallest prime factor function also allowed for the construction of normal numbers. Letting ω(n) stand for the number of distinct prime factors of the positive integer n, we then showed that the concatenation of the successive values of |ω(n)-⌊loglogn⌋| in a fixed base q≥2, as n runs through the integers n≥3, yields a normal number. Here we prove the following. Let q≥2 be a fixed integer. Given an integer n≥n 0 =max(q,3), let N be the unique positive integer satisfying q N ≤n<q N+1 and let h(n,q) stand for the residue modulo q of the number of distinct prime factors of n located in the interval [logN,N]. Setting x N :=e N , we then create a normal number in base q using the concatenation of the numbers h(n,q), as n runs through the integers ≥x n 0 .