We define ψ to be the multiplicative arithmetic function that satisfies ψ(p) = { pα−1(p+ 1), if p 6= 2; pα−1, if p = 2 for all primes p and positive integers α. Let λ(n) be the number of iterations of the function ψ needed for n to reach 2. It follows from a theorem due to White that λ is additive. Following Shapiro’s work on the iterated φ function, we determine bounds for λ. We also use the function λ to partition the set of positive integers into three sets S1, S2, S3 and determine some… CONTINUE READING