Almost perfect number explained

In mathematics, an almost perfect number (sometimes also called slightly defective or least deficient number) is a natural number n such that the sum of all divisors of n (the sum-of-divisors function σ(n)) is equal to 2n − 1, the sum of all proper divisors of n, s(n) = σ(n) − n, then being equal to n − 1. The only known almost perfect numbers are powers of 2 with non-negative exponents . Therefore the only known odd almost perfect number is 2⁰ = 1, and the only known even almost perfect numbers are those of the form 2^k for some positive integer k; 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 six prime factors.^{[1] [2]}

If m is an odd almost perfect number then is a Descartes number.^[3] Moreover if a and b are positive odd integers such that

b +3 <a<\sqrt{m/ 2 } and such that and are both primes, then would be an odd weird number.^[4]

See also

• Perfect number
• Quasiperfect number

Further reading

• Book: Guy, R. K. . Richard K. Guy . Almost Perfect, Quasi-Perfect, Pseudoperfect, Harmonic, Weird, Multiperfect and Hyperperfect Numbers . B2 -->. . 2nd . New York . . 16, 45–53 . 1994 .
• Book: Sándor . József . Mitrinović . Dragoslav S. . Crstici . Borislav . Handbook of number theory I . Dordrecht . . 2006 . 1-4020-4215-9 . 1151.11300 . 110 .
• Book: Sándor . Jozsef . Crstici . Borislav . Handbook of number theory II . Dordrecht . Kluwer Academic . 2004 . 1-4020-2546-7 . 37–38 . 1079.11001 .
• Book: Singh, S. . Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem . registration . New York . Walker . 13 . 1997 . 9780802713315 .

Notes and References

1. Kishore . Masao . Odd integers N with five distinct prime factors for which 2−10⁻¹² < σ(N)/N < 2+10⁻¹² . . 32 . 303–309 . 1978 . 0025-5718 . 0376.10005 . 0485658. 10.2307/2006281. 2006281 .
2. Kishore . Masao . On odd perfect, quasiperfect, and odd almost perfect numbers . . 36 . 583–586 . 1981 . 154 . 0025-5718 . 0472.10007 . 10.2307/2007662. 2007662 . free .
3. Book: Banks . William D. . Güloğlu . Ahmet M. . Nevans . C. Wesley . Saidak . Filip . Descartes numbers . 167–173 . De Koninck . Jean-Marie . Jean-Marie De Koninck . Granville . Andrew . Andrew Granville . Luca . Florian . Anatomy of integers. Based on the CRM workshop, Montreal, Canada, March 13–17, 2006 . Providence, RI . . CRM Proceedings and Lecture Notes . 46 . 2008 . 978-0-8218-4406-9 . 1186.11004 .
4. Melfi . Giuseppe . Giuseppe Melfi . On the conditional infiniteness of primitive weird numbers . . 147 . 508–514 . 2015 . 10.1016/j.jnt.2014.07.024 . free .