Almost perfect number

2020 Mathematics Subject Classification: Primary: 11A [MSN][ZBL]

Slightly defective number or least deficient number

A natural number $n$ such that the sum of all divisors 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 MR0485658. 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