Deficient number: Difference between revisions
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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. |
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. |
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An infinite number of both [[even and odd numbers|even and odd]] deficient numbers exist. For example, all |
An infinite number of both [[even and odd numbers|even and odd]] deficient numbers exist. For example, all odd numbers with one or two distinct prime factors, and all proper [[divisor]]s of deficient or [[perfect number]]s are deficient. |
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Closely related to deficient numbers are [[perfect number]]s with ''σ''(''n'') = 2''n'', and [[abundant number]]s with ''σ''(''n'') > 2''n''. The [[natural number]]s were first classified as either deficient, perfect or abundant by [[Nicomachus]] in his ''Introductio Arithmetica'' (circa 100). |
Closely related to deficient numbers are [[perfect number]]s with ''σ''(''n'') = 2''n'', and [[abundant number]]s with ''σ''(''n'') > 2''n''. The [[natural number]]s were first classified as either deficient, perfect or abundant by [[Nicomachus]] in his ''Introductio Arithmetica'' (circa 100). |
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Revision as of 22:08, 22 December 2009
Divisibility-based sets of integers | ||
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| Overview |  | |
| Factorization forms | ||
| Constrained divisor sums | ||
| With many divisors | ||
| Aliquot sequence-related | ||
| Base-dependent | ||
| Other sets | ||
In mathematics, a deficient number or defective number is a number n for which σ(n) < 2n. Here σ(n) is the sum-of-divisors function: the sum of all positive divisors of n, including n itself. An equivalent definition is that the sum of all proper divisors of the number (divisors other than the number itself) is less than the number. The value 2n − σ(n) is called the deficiency of n.
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 the 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.
An infinite number of both even and odd deficient numbers exist. For example, all odd numbers with one or two distinct prime factors, and all proper divisors of deficient or perfect numbers are deficient.
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).