Divisibility-based sets of integers

Forms of factorization:

Prime number

Composite number

Powerful number

Square-free number

Achilles number

Constrained divisor sums:

Perfect number

Almost perfect number

Quasiperfect number

Multiply perfect number

Hyperperfect number

Superperfect number

Unitary perfect number

Semiperfect number

Primitive semiperfect number

Practical number

Numbers with many divisors:

Abundant number

Highly abundant number

Superabundant number

Colossally abundant number

Highly composite number

Superior highly composite number

Other:

Untouchable number

Deficient number

Weird number

Amicable number

Friendly number

Sociable number

Solitary number

Sublime number

Harmonic divisor number

Frugal number

Equidigital number

Extravagant number

See also:

Divisor function

Divisor

Prime factor

Factorization

This box: view • talk • edit

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. The only known odd almost perfect number is 2⁰ = 1, and the only even almost perfect numbers known 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.

[edit] External links

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

[edit] 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.
• Sloane, N. J. A., Sequence A000079/M1129 in "The On-Line Encyclopedia of Integer Sequences."

This number theory-related article is a stub. You can help Wikipedia by expanding it. v • d • e