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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.

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]


Original courtesy of Wikipedia: http://en.wikipedia.org/wiki/Deficient_number — Please support Wikipedia.
This page uses Creative Commons Licensed content from Wikipedia. A portion of the proceeds from advertising on Digplanet goes to supporting Wikipedia.

18 news items

 
Main Line
Thu, 17 Jul 2014 20:29:00 -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 ...
 
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 ...
 
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 ...

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.

Washington Times (blog)

Washington Times (blog)
Mon, 22 Jul 2013 04:23:18 -0700

$257,000 to “prepare graduates from the doctoral program at the University of Texas Medical Branch School of Nursing to assume faculty roles in schools for nursing with a deficient number of doctoral level faculty.” 13. $1.4 million to buy buses for ...
 
Shelbyville Times-Gazette
Tue, 14 May 2013 08:12:47 -0700

"There were a deficient number of smoke detectors, several other hazards there and the caregiver couldn't take care of them. "The other hazards included light fixtures with exposed wires hanging from the ceiling and smoking inside the residence by the ...

The Hindu

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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