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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

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 ...

Bemidji Pioneer

Bemidji Pioneer
Sat, 25 Oct 2014 21:33:41 -0700

BEMIDJI — Weather in Bemidji last week was ideal for camping. In fact, one man was camping in the woods last Monday — when Beltrami County Sheriff's Office deputies responded to a suspicious person call. The man was staying in a tent until his ...
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 ...
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 ...
GeekDad (blog)
Tue, 15 Apr 2014 03:06:01 -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 ...


Sat, 07 Jun 2014 19:03:45 -0700

“It is a preliminary inquiry to initially determine the cause of deaths. The team pored over documents apart from questioning certain relevant people and concludes that a deficient number of doctors and paramedics were present in the facility,” he said.

New York Daily News

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

When it comes to improving its infrastructure, America seems to be on a bridge to nowhere. One out of every nine bridges in the United States—a grand total of 66,503 bridges altogether—has been found to have one or more significant defects, according ...

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