Deficient number
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 (sequence A005100 in the OEIS) are
As an example, consider the number 21. Its divisors are 1, 3, 7 and 21, whose 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 prime numbers, all prime powers 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).