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Powerfree sums of proper divisors. (English) Zbl 1496.11124

Let \(n\geqslant 2\), and let \(s(n)=\sum_{d|n,d<n}d\) denote the sum of the proper divisors of \(n\). A natural number \(n\geqslant 2\) is called \(k\)-free if \(n\) is not divisible by the \(k\)-th power of an integer large than \(1\).
Among others related results, the authors of the present paper show that \(n\) is \(k\)-free if and only if \(s(n)\) is \(k\)-free in the case \(k\geqslant 4\).

MSC:

11N37 Asymptotic results on arithmetic functions
11A25 Arithmetic functions; related numbers; inversion formulas
11N64 Other results on the distribution of values or the characterization of arithmetic functions
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[16] Akash Singha Roy ESIC Staff Quarters No.: D2
[17] Sterling Road, Nungambakkam Chennai 600034
[18] Tamil Nadu, India E-mail: akash01s.roy@gmail.com
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