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In number theory, a deficient or deficient 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.


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.


  • 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]


  1. ^ Sándor et al (2006) p.108

External links[edit]

Original courtesy of Wikipedia: http://en.wikipedia.org/wiki/Deficient_number — Please support Wikipedia.
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21 news items

The Patriot-News - PennLive.com
Mon, 22 Sep 2014 13:53:48 -0700

"It's going to knock down this structurally deficient number that we've had hanging over our heads for a long time." Work on the bridge replacements is expected to begin next year and 99 percent of the 558 bridges are to be finished by August 31, 2018.
Main Line
Mon, 21 Jul 2014 04:09:52 -0700

“We've made great strides in reducing that structurally deficient number in the last few years,” she said. “That doesn't mean that each one is going to fall into the water tomorrow,” said Rapoport. “It could be that a couple of them do, similar to that ...

Hurriyet Daily News

Hurriyet Daily News
Thu, 11 Sep 2014 02:22:50 -0700

This, sadly, is a deficient number,” Aslan said. “According to the data from the ministry, 22,000 people chose Kurdish as an elective course this year across Turkey. There will be one Kurdish teacher for every 1,100 students if only 20 teachers are ...
GeekDad (blog)
Tue, 15 Apr 2014 03:06:01 -0700

In the field of Number Theory, you can classify a given counting number based on how it compares to the sum of its proper divisors. If a number is exactly equal to the sum of its proper divisors, it is a Perfect Number. These are the least common, and ...
Arkansas News
Mon, 21 Jul 2014 17:00:00 -0700

Give Arkansas a Raise, a group seeking to place on the ballot a proposed initiated act to raise the state minimum wage, also submitted signatures to Martin's office on July 7 and also has been given 30 days to make up for a deficient number of valid ...
Times of India
Fri, 17 Oct 2014 13:29:15 -0700

The fee is stringent when it comes to parking. An amount of Rs 5 lakh is levied as regularization fee for every deficient number. "This is one factor which every builder has to keep in mind while applying for levying parking fee. If the building has ...
Edison Sentinel
Tue, 18 Mar 2014 07:55:07 -0700

Grygiel said the variances needed for the project — for the deficient number and size of parking spaces, the size of the front yard setback, the building height and several waivers — will not outweigh its positives. “This is a textbook example of ...

New York Daily News

New York Daily News
Fri, 02 Aug 2013 17:28:37 -0700

One in nine U.S. bridges are structurally deficient, but few Washington politicians seem willing to confront the problem. BY David Knowles. NEW YORK DAILY NEWS. Published: Friday, August 2, 2013, 8:26 PM. Updated: Friday, August 2, 2013, 8:26 PM.

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