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Thursday, January 22, 2015

**Abstract:** 1. Let $f$ be the arithmetic function so that $f(p^k) = p^{k-1}(p+1)$. THM: For any integer $m$, there exists $N$ so that if $n \ge N$, then the $n$-th iterate $f^n(m) = 2^{n+a(m)}3^{b(m)}.$ Similar results hold if ``1" is replaced by ``d".

2. Let $C$ denote the standard Cantor set. Since $C \subset [0,1/3] \cup [2/3,1]$, if $x \in (1/3,4/9)$, then $x$ is not a product of two members of $C$. THM: If $x \in [0,1]$, then $x$ is a product of three members of $C$. This work and its generalizations are joint with Jayadev Athreya and Jeremy Tyson.