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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)', 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.

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.
• There exists at least one deficient number in the interval $[n, n + (\log n)^2]$ for all sufficiently large n.[1]

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).

References

1. ^ Sándor et al (2006) p.108
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 The Hindu 'We are fighting a war against polio without a General' The Hindu Wed, 12 Jun 2013 12:16:28 -0700 Therefore, it was the combination of deficient number of doses plus the unlucky event of getting infected with a vaccine-derived virus. Not every child who gets infected with VDPV will become paralysed — so twice unlucky. Does the emergence of VDPV ...
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