Constructing normal numbers using residues of selective prime factors of integers
Jean–Marie De Koninck, Imre Kátai · Annales Universitatis Scientiarum Budapestinensis de Rolando Eötvös Nominatae Sectio computatorica · 2014
Given an integer N ≥ 1, for each integer n ∈ JN := [e N , e N +1 ), let qN (n) be the smallest prime factor of n which is larger than N ; if no such prime factor exists, set qN (n) = 1.Fix an integer Q ≥ 3 and consider the function f (n) = fQ(n) defined by f (n) = ℓ if n ≡ ℓ (mod Q) with (ℓ, Q) = = 1 and by f (n) = Λ otherwise, where Λ stands for the empty word.Then consider the sequence (κThen, for each integer N ≥ 1, consider the concanetation of the numbers κ(1), κ(2), . .., that is define θN := Concat(κ(n) : n ∈ JN ).Then, set αQ := Concat(θN : N = 1, 2, 3, . ..).Finally, let BQ = {ℓ1, ℓ2, . . ., ℓ ϕ(Q) } be the set of reduced residues modulo Q, where ϕ stands for the Euler function.We show that αQ is a normal sequence over BQ.