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

Abstract

We define $\overline{\psi }$ 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 $\alpha$. Let $\lambda (n)$ be the number of iterations of the function $\overline{\psi }$ 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.

13 pages, 0 figures