Journal Article An arithmetic function arising from the Dedekind ψ function

Defant, Colin

59 ( 1 )  , pp.81 - 92 , 2017-01 , Department of Mathematics, Faculty of Science, Okayama University
ISSN:0030-1566
NCID:AA00723502
Description
We define &psi;&oline; to be the multiplicative arithmetic function that satisfies<br><img src="http://www.lib.okayama-u.ac.jp/www/mjou/mjou_59_81.png"><br>for all primes <i>p</i> and positive integers &alpha;. Let <i>&lambda;(n)</i> be the number of iterations of the function <i>&psi;&oline;</i> needed for <i>n</i> to reach 2. It follows from a theorem due to White that <i>&lambda;</i> is additive. Following Shapiro's work on the iterated <i>&phi;</i> function, we determine bounds for <i>&lambda;</i>. We also use the function <i>&lambda;</i> to partition the set of positive integers into three sets <i>S<sub>1</sub>, S<sub>2</sub>, S<sub>3</sub></i> and determine some properties of these sets.
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