Almost perfect number

In mathematics, an almost perfect number (sometimes also called slightly defective number) is a natural number n such that the sum of all divisors of n (the divisor 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 (sequence A000079 in OEIS). 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 number k; however, it has not been shown that all almost perfect numbers are of this form. Almost perfect numbers are also known as least deficient numbers.

References

• Guy, R. K., Almost Perfect, Quasi-Perfect, Pseudoperfect, Harmonic, Weird, Multiperfect and Hyperperfect Numbers. §B2 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 16 and 45-53, 1994.
• Singh, S., Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem. New York: Walker, p. 13, 1997.

External links

• Weisstein, Eric W., "Almost perfect number" from MathWorld.