Perfect Abundant and Deficient Numbers

Math Algebra • Numbers

Written by STEM Calculators Team Published May 15, 2026 Updated May 15, 2026

Classify a positive integer as perfect, abundant, or deficient by comparing the number with the sum of its proper divisors.

s(n) = sum of proper divisors Perfect: s(n) = n Abundant: s(n) > n Deficient: s(n) < n

Number input

Positive integer n

Divisor display

Step detail

Supported range: \(1\le n\le 10^8\). Proper divisors are positive divisors less than \(n\). The number \(1\) is deficient because it has no positive proper divisors, so \(s(1)=0\).

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Theory – Perfect, abundant, and deficient numbers

Perfect, abundant, and deficient numbers are classified by comparing a number \(n\) with the sum of its positive proper divisors. This sum is often called the aliquot sum and is written \(s(n)\).

1. Divisors and proper divisors

A positive integer \(d\) is a divisor of \(n\) if \(d\) divides \(n\) exactly:

\[ \begin{aligned} d\mid n \quad\Longleftrightarrow\quad n=dq \end{aligned} \]

for some integer \(q\).

A proper divisor is a positive divisor that is smaller than \(n\). Therefore, \(n\) itself is not counted as a proper divisor.

\[ \begin{aligned} D(n) &= \{d:\ d\mid n,\ 1\le d2. Sum of proper divisors

The sum of proper divisors is

\[ \begin{aligned} s(n) &= \sum_{d\in D(n)} d. \end{aligned} \]

This means we list all proper divisors and add them.

3. Perfect numbers

A number is perfect if the sum of its proper divisors is exactly equal to the number:

\[ \begin{aligned} s(n)&=n. \end{aligned} \]

Example:

\[ \begin{aligned} D(28)&=\{1,2,4,7,14\},\\ s(28)&=1+2+4+7+14\\ &=28. \end{aligned} \]

Since \(s(28)=28\), the number \(28\) is perfect.

4. Abundant numbers

A number is abundant if the sum of its proper divisors is greater than the number:

\[ \begin{aligned} s(n)&>n. \end{aligned} \]

Example:

\[ \begin{aligned} D(12)&=\{1,2,3,4,6\},\\ s(12)&=1+2+3+4+6\\ &=16. \end{aligned} \]

Since \(16>12\), the number \(12\) is abundant.

5. Deficient numbers

A number is deficient if the sum of its proper divisors is less than the number:

\[ \begin{aligned} s(n)& Example:

\[ \begin{aligned} D(15)&=\{1,3,5\},\\ s(15)&=1+3+5\\ &=9. \end{aligned} \]

Since \(9<15\), the number \(15\) is deficient.

6. The special case \(1\)

The number \(1\) has no positive proper divisors. Therefore:

\[ \begin{aligned} D(1)&=\varnothing,\\ s(1)&=0. \end{aligned} \]

Since \(0<1\), the number \(1\) is deficient.

7. Efficient divisor search

Divisors come in pairs. If \(d\mid n\), then \(n/d\) is also a divisor. For example, for \(28\):

\[ \begin{aligned} 1\cdot28&=28,\\ 2\cdot14&=28,\\ 4\cdot7&=28. \end{aligned} \]

Because of this pairing, it is enough to test possible divisors up to \(\sqrt n\).

\[ \begin{aligned} d\le\sqrt n. \end{aligned} \]

8. Worked example: \(28\)

First list the proper divisors:

\[ \begin{aligned} D(28)&=\{1,2,4,7,14\}. \end{aligned} \]

Now add them:

\[ \begin{aligned} s(28) &=1+2+4+7+14\\ &=28. \end{aligned} \]

Compare \(s(28)\) with \(28\):

\[ \begin{aligned} s(28)&=28. \end{aligned} \]

Therefore:

\[ \begin{aligned} \boxed{28\text{ is perfect}.} \end{aligned} \]

9. Aliquot difference

The aliquot difference is

\[ \begin{aligned} s(n)-n. \end{aligned} \]

It summarizes the classification:

\[ \begin{aligned} s(n)-n&=0 &&\Rightarrow \text{perfect},\\ s(n)-n&>0 &&\Rightarrow \text{abundant},\\ s(n)-n&<0 &&\Rightarrow \text{deficient}. \end{aligned} \]

10. Abundancy ratio

Another useful comparison is the ratio

\[ \begin{aligned} \frac{s(n)}{n}. \end{aligned} \]

This ratio is equal to \(1\) for perfect numbers, greater than \(1\) for abundant numbers, and less than \(1\) for deficient numbers.

11. Sum of all divisors

The sum of all positive divisors, including \(n\), is usually written \(\sigma(n)\). Since \(s(n)\) excludes \(n\), we have:

\[ \begin{aligned} \sigma(n)&=s(n)+n. \end{aligned} \]

Therefore:

\[ \begin{aligned} s(n)&=\sigma(n)-n. \end{aligned} \]

12. Common examples

Number	Proper divisors	Divisor sum	Classification
6	1, 2, 3	s(6) = 6	Perfect
28	1, 2, 4, 7, 14	s(28) = 28	Perfect
12	1, 2, 3, 4, 6	s(12) = 16	Abundant
18	1, 2, 3, 6, 9	s(18) = 21	Abundant
15	1, 3, 5	s(15) = 9	Deficient
1	none	s(1) = 0	Deficient

13. Formula summary

The table below uses plain text formulas to prevent raw LaTeX from appearing inside table cells on pages where MathJax does not process table content reliably.

Concept	Formula or test	Meaning
Proper divisors	D(n) = { d : d divides n and 1 ≤ d < n }	Positive divisors smaller than n
Aliquot sum	s(n) = sum of all d in D(n)	Sum of proper divisors
Perfect	s(n) = n	Proper divisors add exactly to n
Abundant	s(n) > n	Proper divisors add to more than n
Deficient	s(n) < n	Proper divisors add to less than n
Aliquot difference	s(n) − n	Zero, positive, or negative classification check
All-divisor sum	σ(n) = s(n) + n	Includes n itself

14. Common mistakes

Including \(n\) itself in the proper divisor sum.
Forgetting that \(1\) is a proper divisor of every integer greater than \(1\).
Thinking prime numbers are perfect; every prime number is deficient because its only proper divisor is \(1\).
Forgetting that \(1\) has no positive proper divisors.
Stopping the divisor list too early and missing paired divisors above \(\sqrt n\).

Key idea: compare \(s(n)\) with \(n\). Equal means perfect, greater means abundant, and less means deficient.

Frequently Asked Questions

What is a proper divisor?

A proper divisor of n is a positive divisor that is smaller than n. The number n itself is not included.

What is s(n)?

s(n) is the sum of all positive proper divisors of n.

How do you know whether a number is perfect?

A number is perfect if the sum of its proper divisors equals the number itself. For example, 28 is perfect because 1 + 2 + 4 + 7 + 14 = 28.

How do you know whether a number is abundant?

A number is abundant if the sum of its proper divisors is greater than the number. For example, 12 is abundant because 1 + 2 + 3 + 4 + 6 = 16, and 16 is greater than 12.

How do you know whether a number is deficient?

A number is deficient if the sum of its proper divisors is less than the number. For example, 15 is deficient because 1 + 3 + 5 = 9, and 9 is less than 15.

Is 1 perfect, abundant, or deficient?

The number 1 is deficient because it has no positive proper divisors, so s(1) = 0, and 0 is less than 1.

Are prime numbers perfect?

No. Every prime number has only one positive proper divisor, 1, so every prime number is deficient.

Why can divisors be found by checking only up to sqrt(n)?

Divisors come in pairs. If d divides n, then n/d is also a divisor. One member of each divisor pair is less than or equal to sqrt(n), so checking up to sqrt(n) is enough.

What is the aliquot difference?

The aliquot difference is s(n) - n. It is zero for perfect numbers, positive for abundant numbers, and negative for deficient numbers.

What is sigma(n)?

sigma(n), written σ(n), is the sum of all positive divisors of n, including n itself. It satisfies σ(n) = s(n) + n.

Perfect Abundant and Deficient Numbers

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