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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.

Examples[edit]

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[edit]

  • 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[edit]

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[edit]

References[edit]

  1. ^ Sándor et al (2006) p.108

External links[edit]



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2 news items

 
GeekDad (blog)
Tue, 15 Apr 2014 02:56:15 -0700

If a number is greater than the sum of its proper divisors, it is a Deficient Number. Most odd numbers are Deficient, as are all the prime numbers. The first few Deficient Numbers are 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, and 17. If a number ...
 
GeekDad (blog)
Wed, 09 Apr 2014 03:09:39 -0700

The 10th lowest perfect number has over 50 digits, and the 16th lowest over 1000 digits. If a number is greater than the sum of its proper divisors, it is a Deficient Number. Most odd numbers are Deficient, as are all the prime numbers. The first few ...
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