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Demonstration, with Cuisenaire rods, of the deficiency of the number 8

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.
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Wed, 13 Apr 2016 02:31:07 -0700

Ten years ago, Army Aviation was reeling from a deficient number of helicopters to effectively support the Army's COIN efforts in FATA, but ten years from now, it could be a potentially be a comparatively much larger and multi-layered force, one with a ...

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Mon, 29 Feb 2016 04:00:00 -0800

Smock Bridge is in need of reconstruction with crumbling railings, potholes and rusting support beams. Smock Bridge is scheduled to be rehabilitated during this construction season, but Pennsylvania is second worst both in number of structurally ...

Sunbury Daily Item

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Thu, 18 Feb 2016 20:02:49 -0800

The state needs to repair or replace roughly 300 bridges a year to ensure the structurally deficient number is not rising, Waters-Trasatt said. “Bridges are the 800-pound gorilla,” said Madden. “This is nothing against the state police, they need money ...

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That's it. We'll never know if LeBron scratched the play and called his own, three-deficient number (28%). Seriously. Is there anybody on the team with a lower three-point shooting percentage? Anyone who'd be a worse choice to take a game winning three?
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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 ...
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While not disagreeing on the need for more affordable housing, she said (as others have said at prior meetings) that there are safety issues (particularly when it snows) and a deficient number of visitor parking spaces (which would lead to parking on ...
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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 ...

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