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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 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]
 ^ Sándor et al (2006) p.108
External links[edit]
Divisibilitybased sets of integers


Overview 



Factorization forms 


Constrained divisor sums 


With many divisors 


Aliquot sequencerelated 


Other sets 


Orléans Star 
Orléans Star
Fri, 12 Jun 2015 14:00:00 0700
... a thirdpart engineering firm, explained that there were several "alarming findings" including structural challenges, sagging roof lines, roof leaks, outdated electrical panel boards, a lack of emergency systems/lighting, and a deficient number of ...
 
TribuneReview
Wed, 20 May 2015 19:40:16 0700
“More money would help us get that structurally deficient number down, which we're trying to do.” Bridge inspections occur annually. Then the county, in conjunction with PennDOT, prioritizes repairs and preventative maintenance. The sheer number of ...
 580 CFRA Radio 
580 CFRA Radio
Fri, 05 Jun 2015 05:30:12 0700
A December 2014 building audit found the building had structural challenges, sagging roof lines, leaks, outdated electrical boards, a lack of emergency systems and lights and a deficient number of exhaust fans. Engineers also found mouse droppings.
 
The PatriotNews  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.
 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 ...
 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 ...
 
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 ...
 
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 ...

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