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 is 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. Odd integers N with five distinct prime factors for which 2−10−12 < σ(N)/N < 2+10−12. Mathematics of Computation 32 (1978). p. 303–309. doi:10.2307/2006281.
- ↑ Kishore, Masao. On odd perfect, quasiperfect, and odd almost perfect numbers. Mathematics of Computation 36 (154) (1981). p. 583–586. doi:10.2307/2007662.
Further reading
- Guy, R. K.. Unsolved Problems in Number Theory (1994). New York: Springer-Verlag. p. 16, 45–53.
- Handbook of number theory I (2006). Dordrecht: Springer-Verlag. p. 110. ISBN 1-4020-4215-9.
- Handbook of number theory II (2004). Dordrecht: Kluwer Academic. p. 37–38. ISBN 1-4020-2546-7.
- Singh, S.. Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem (1997). New York: Walker. p. 13. ISBN 9780802713315.