Almost perfect number
A visual example to show that 8 is almost perfect and deficient.
In math, an almost perfect number (also called slightly defective or least deficient number) is a type of natural number n. The sum of n's divisors must be equal to 2n − 1. Every known almost perfect number is a power of 2 and has non-negative exponents (sequence [{{fullurl:OEIS:{{{id}}}}} {{{id}}}] in OEIS).
Examples
For example, the divisors of 32 are 1, 2, 4, 8, 16 and 32. The sum of those is 63. 32 ⋅ 2 - 1 is 63. This makes 32 an almost perfect number.
Odd numbers
The only known odd almost perfect number 1. An odd almost perfect number that's not 1 is possible. It would, however, have to have six prime factors.[1][2]
Almost Perfect Number Media
Demonstration, with Cuisenaire rods, that the number 8 is almost perfect, and deficient.
References
- ↑ Kishore, Masao (1978). "Odd integers N with five distinct prime factors for which 2−10−12 < σ(N)/N < 2+10−12" (PDF). Mathematics of Computation. 32: 303–309. doi:10.2307/2006281. ISSN 0025-5718. JSTOR 2006281. MR 0485658. Zbl 0376.10005.
- ↑ Kishore, Masao (1981). "On odd perfect, quasiperfect, and odd almost perfect numbers". Mathematics of Computation. 36 (154): 583–586. doi:10.2307/2007662. ISSN 0025-5718. JSTOR 2007662. Zbl 0472.10007.
Further reading
- Guy, R. K. (1994). "Almost Perfect, Quasi-Perfect, Pseudoperfect, Harmonic, Weird, Multiperfect and Hyperperfect Numbers". Unsolved Problems in Number Theory (2nd ed.). New York: Springer-Verlag. pp. 16, 45–53.
- Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, eds. (2006). Handbook of number theory I. Dordrecht: Springer-Verlag. p. 110. ISBN 1-4020-4215-9. Zbl 1151.11300.
- Sándor, Jozsef; Crstici, Borislav, eds. (2004). Handbook of number theory II. Dordrecht: Kluwer Academic. pp. 37–38. ISBN 1-4020-2546-7. Zbl 1079.11001.
- Singh, S. (1997). Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem. New York: Walker. p. 13. ISBN 9780802713315.