The Number of Prime Factors Function on Shifted Primes and Normal Numbers

In a series of papers, we have 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 show that the concatenation of the successive values of | ω ( n ) − ⌊ log log n ⌋ | $$\vert \omega (n) -\lfloor \log \log n\rfloor \vert $$ , as n runs through the integers n ≥ 3, yields a normal number in any given basis q ≥ 2. We show that the same result holds if we consider the concatenation of the successive values of | ω ( p + 1 ) − ⌊ log log ( p + 1 ) ⌋ | $$\vert \omega (p + 1) -\lfloor \log \log (p + 1)\rfloor \vert $$ , as p runs through the prime numbers.