An Arithmetic Function Arising from the Dedekind $\psi$ Function

We define $\bar{\psi }$ to be the multiplicative arithemtic function that satisfies $$\bar{\psi }(p^{\alpha })=\{\begin{array}{ll}{p}^{\alpha -1}(p+1),& ifp\mathrm{\ne }2;\\ p^{\alpha -1},& ifp=2\end{array}$$ for all primes $p$ and positive integers $\alpha$. Let $\lambda (n)$ be the number of iterations of the function $\bar{\psi }$ needed for $n$ to reach $2$. It follows from a theorem due to White that $\lambda$ is additive. Following Shapiro’s work on the iterated $\phi$ function, we determine bounds for $\lambda$. We also use the function $\lambda$ to partition the set of positive integers into three sets $S_1,S_2,S_3$ and determine some properties of these sets.