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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, etc. 11N25 Distribution of integers with specified multiplicative constraints
##### Keywords:
sum of divisors; (almost) perfect numbers