An Arithmetic Function Arising from the Dedekind $ψ$ Function

Abstract

We define $\overlineψ$ to be the multiplicative arithemtic function that satisfies \[\overlineψ(p^α)=\begin{cases} p^{α-1}(p+1), & \mbox{if } p\neq 2; \\ p^{α-1}, & \mbox{if } p=2 \end{cases}\] for all primes $p$ and positive integers $α$. Let $λ(n)$ be the number of iterations of the function $\overlineψ$ 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 $S_1,S_2,S_3$ and determine some properties of these sets.