## 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

### Keywords:

sum of divisors; (almost) perfect numbers