View source for Descartes number

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A number which is close to being a [[perfect number]].  They are named for René Descartes who observed that the number 

$$D= 198585576189 = 3^2⋅7^2⋅11^2⋅13^2⋅22021 $$

would be an odd perfect number if only 22021 were a [[prime number]], since the [[Sum of divisors|sum-of-divisors function]] for $D$ satisfies

$$\sigma(D) = (3^2+3+1)\cdot(7^2+7+1)\cdot(11^2+11+1)\cdot(13^3+13+1)\cdot(22021+1) \ . $$

A Descartes number is defined as an odd number $n = m p$ where $m$ and $p$ are coprime and $2n = \sigma(m)\cdot(p+1)$.  The example given is the only one currently known.

If $m$ is an odd [[almost perfect number]], that is, $\sigma(m) = 2m-1$, then $m(2m−1)$ is a Descartes number.

==References==
* Banks, William D.; Güloğlu, Ahmet M.; Nevans, C. Wesley; Saidak, Filip. "Descartes numbers". In De Koninck, Jean-Marie; Granville, Andrew; Luca, Florian (edd). ''Anatomy of integers. Based on the CRM workshop, Montreal, Canada, March 13--17, 2006''. CRM Proceedings and Lecture Notes '''46''' Providence, RI: American Mathematical Society (2008) pp. 167–173. {{ISBN|978-0-8218-4406-9}}. {{ZBL|1186.11004}}.