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Descartes numbers. (English) Zbl 1186.11004

De Koninck, Jean-Marie (ed.) et al., Anatomy of integers. Based on the CRM workshop, Montreal, Canada, March 13–17, 2006. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4406-9/pbk). CRM Proceedings and Lecture Notes 46, 167-173 (2008).
Let \(\sigma\) denote the sum of divisors function. The authors call an integer \(n\) a Descartes number if \(n\) is odd and if \(n=km\) for two integers \(k,m>1\) such that \[ \sigma(k)(m+1)=2n. \] They prove:
Theorem 1. If \(n\) is a cube-free Descartes number which is not divisible by \(3\), then \(n=k\sigma(k)\) for some odd almost perfect number \(k\), and \(n\) has more than one million distinct prime divisors.
Theorem 2. The number \(3^27^211^213^222021\) is the only cube-free Descartes number with fewer than seven distinct prime divisors.
For the entire collection see [Zbl 1142.11002].

MSC:

11A25 Arithmetic functions; related numbers; inversion formulas
11N25 Distribution of integers with specified multiplicative constraints
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