Hypercyclic and chaotic convolution operators associated with the Dunkl operator on <Emphasis Type=”Bold”>C</Emphasis>

We show that there exist infinitely many positive integers m not of the form <Emphasis Type=”Italic”>n</Emphasis>-&ohgr;(<Emphasis Type=”Italic”>n</Emphasis>) for any positive integer <Emphasis Type=”Italic”>n</Emphasis>. Here, &ohgr;(<Emphasis Type=”Italic”>n</Emphasis>) stands for the number of distinct prime factors of <Emphasis Type=”Italic”>n</Emphasis>. A similar result holds with &ohgr;(<Emphasis Type=”Italic”>n</Emphasis>) replaced by the total number of prime factors of <Emphasis Type=”Italic”>n</Emphasis> (counting multiplicities), or by the number of divisors of <Emphasis Type=”Italic”>n</Emphasis>.