View source for Almost perfect number

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''Slightly defective number'' or ''least deficient number''

A [[natural number]] $n$ such that the [[sum]] of all [[divisor]]s of $n$ (the [[sum of divisors]] function $\sigma(n)$) is equal to $2n − 1$.  The only known almost perfect numbers are the powers of 2 with non-negative exponents; however, it has not been shown that all almost perfect numbers are of this form.  It is known that an odd almost perfect number greater than 1 would have at least 6 prime factors.

If $m$ is an odd almost perfect number then $m(2m-1)$ is a [[Descartes number]].

==References==
* Kishore, Masao. "Odd integers N with five distinct prime factors for which $2−10^{−12} < \sigma(N)/N < 2+10^{−12}$". Mathematics of Computation '''32''' (1978) 303–309. {{DOI|10.2307/2006281}} {{MR|0485658}}. {{ZBL|0376.10005}} 
* Kishore, Masao. "On odd perfect, quasiperfect, and odd almost perfect numbers". Mathematics of Computation '''36''' (1981) 583–586. {{ZBL|0472.10007}} 
* 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}} 
* Guy, R. K. . "Almost Perfect, Quasi-Perfect, Pseudoperfect, Harmonic, Weird, Multiperfect and Hyperperfect Numbers". Unsolved Problems in Number Theory (2nd ed.). New York: Springer-Verlag (1994). pp. 16, 45–53 
* Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, edd. (2006). Handbook of number theory I. Dordrecht: Springer-Verlag (2006). p.110. {{ISBN|1-4020-4215-9}} {{ZBL|1151.11300}} 
* Sándor, Jozsef; Crstici, Borislav, edd. Handbook of number theory II. Dordrecht: Kluwer Academic (2004). pp.37–38. {{ISBN|1-4020-2546-7}}  {{ZBL|1079.11001}}