- Deficient number
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In number theory, a deficient number or defective number is a number n for which the sum of divisors σ(n)<2n, or, equivalently, the sum of proper divisors (or aliquot sum) s(n)<n. The value 2n − σ(n) (or n − s(n)) is called the number's deficiency.
Contents
Examples
The first few deficient numbers are:
- 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, … (sequence A005100 in OEIS)
As an example, consider the number 21. Its divisors are 1, 3, 7 and 21, and their sum is 32. Because 32 is less than 2 × 21, the number 21 is deficient. Its deficiency is 2 × 21 − 32 = 10.
Properties
- An infinite number of both even and odd deficient numbers exist
- All odd numbers with one or two distinct prime factors are deficient
- All proper divisors of deficient or perfect numbers are deficient.
Related Concepts
Closely related to deficient numbers are perfect numbers with σ(n) = 2n, and abundant numbers with σ(n) > 2n. The natural numbers were first classified as either deficient, perfect or abundant by Nicomachus in his Introductio Arithmetica (circa 100).
See also
External links
- The Prime Glossary: Deficient number
- Weisstein, Eric W., "Deficient Number" from MathWorld.
- deficient number at PlanetMath.
Divisibility-based sets of integers Overview 
Forms of factorization Constrained divisor sums Numbers with many divisors Abundant number · Primitive abundant number · Highly abundant number · Superabundant number · Colossally abundant number · Highly composite number · Superior highly composite number · Weird numberNumbers related
to aliquot sequencesOther Deficient number · Friendly number · Solitary number · Sublime number · Harmonic divisor number · Frugal number · Equidigital number · Extravagant numberCategories:- Divisor function
- Integer sequences
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