thumb|Demonstration, with [[Cuisenaire rods, that the number 8 is almost perfect, and deficient.]]
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 2<sup>0</sup> = 1, and the only known even almost perfect numbers are those of the form 2<sup>k</sup> 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.
If m is an odd almost perfect number then is a Descartes number. Moreover if a and b are positive odd integers such that <math>b+3<a<\sqrt{m/2}</math> and such that and are both primes, then would be an odd weird number.
See also
- Perfect number
- Quasiperfect number