On a Problem in Elementary Number Theory

Let A be any positive integer and let Φ(x, A) denote the number of positive integers which are prime to A and less than or equal to x. The object of this note is to prove a result about the upper and lower bounds of e₁, e₂, e₃,..., where

e_n = e_A,n = Φ(n,A)-n Φ(A)/A, n = 1, 2, 3 , . . .,

and Φ(A) stands for Φ(A, A).