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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, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, ... (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.
This page uses Creative Commons Licensed content from Wikipedia. A portion of the proceeds from advertising on Digplanet goes to supporting Wikipedia.

21 news items

Insurance Journal
Thu, 19 Nov 2015 09:48:45 -0800

If a licensee does not complete the 24 hours of CE before the expiration date of the license, the licensee will have 90 days to complete the deficient number of hours and pay a fine of $50 per deficient hour. If these two conditions are not met within ...
Loudoun Times-Mirror
Wed, 30 Sep 2015 08:14:26 -0700

The second brewery in Loudoun with a hairless man on its logo (the other is Belly Love Brewery), Bald Guy developed as a response to what they see as a “deficient number of places to relax, chat with friends and neighbors, and not feel like we're at a ...

Reboot Illinois

Reboot Illinois
Tue, 28 Apr 2015 10:31:41 -0700

More than 134 million crossings occur each day on Illinois' 26,588 bridges, which span nearly 13 million square meters. Of those 26,588 state and federal bridges, 2,216 are structurally deficient, meaning one or more key bridge components — deck, ...

Huffington Post

Huffington Post
Wed, 29 Apr 2015 12:44:13 -0700

Illinoisans cross the 13 million square miles of Illinois' 26,588 bridges more than 134 million times every day. Of those 26,588 state and federal bridges, 2,216 are structurally deficient, meaning one or more key bridge components (i.e., deck ...

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.

About.com: Health

About.com: Health
Thu, 12 Feb 2015 09:23:14 -0800

The first limitation is power or deficient number of participants (small sample sizes). We must keep in mind, however, that sample sizes of even a few hundred participants can yield statistically significant results. Moreover, the global prevalence or ...

GeekDad (blog)

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 ...

The Patriot-News - PennLive.com

The Patriot-News - PennLive.com
Mon, 22 Sep 2014 13:48:45 -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.

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