Sum of Divisors Calculator – Find σ(n), Divisors & Factorization

What Is the Sum of Divisors?

The sum of divisors of a positive integer is the sum of all its positive divisors, including 1 and the number itself.

In number theory, this function is called the divisor-sum function and is written as \(\sigma(n)\).

For example, the positive divisors of 12 are:

\[ 1,\;2,\;3,\;4,\;6,\;12 \]

Therefore:

\[ \sigma(12) = 1+2+3+4+6+12 = 28 \]

This calculator automatically finds the divisors and calculates their sum, so you do not have to find them manually.

Sum of Divisors Formula

If the prime factorization of \(n\) is known, the sum of divisors can be calculated without listing every divisor.

Suppose:

\[ n = p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k} \]

Then:

\[ \boxed{ \sigma(n) = \prod_{i=1}^{k} \frac{p_i^{a_i+1}-1}{p_i-1} } \]

This is equivalent to multiplying the geometric sums:

\[ \sigma(n) = (1+p_1+p_1^2+\cdots+p_1^{a_1}) \cdots (1+p_k+p_k^2+\cdots+p_k^{a_k}) \]

Sum of Divisors Examples

Example 1: Sum of divisors of 6

The positive divisors of 6 are:

1, 2, 3, 6

Therefore:

\[ \sigma(6) = 1+2+3+6 = 12 \]

The proper divisors are 1, 2 and 3:

\[ 1+2+3=6 \]

Therefore, 6 is a perfect number.

Example 2: Sum of divisors of 12

The positive divisors are:

1, 2, 3, 4, 6, 12

\[ \sigma(12) = 1+2+3+4+6+12 = 28 \]

The proper divisors sum to:

\[ 1+2+3+4+6=16 \]

Since 16 is greater than 12, 12 is an abundant number.

Example 3: Sum of divisors of a prime number

A prime number has exactly two positive divisors: 1 and the number itself.

Therefore, for a prime \(p\):

\[ \sigma(p)=1+p \]

For example:

\[ \sigma(13)=1+13=14 \]

Example 4: Sum of divisors of 28

The positive divisors of 28 are:

1, 2, 4, 7, 14, 28

\[ \sigma(28) = 1+2+4+7+14+28 = 56 \]

Its proper divisors sum to:

\[ 1+2+4+7+14=28 \]

Therefore, 28 is also a perfect number.

Related Divisor Concepts

Divisor Count \( \tau(n) \)

The divisor-counting function, commonly written as \(\tau(n)\) or \(d(n)\), gives the number of positive divisors of \(n\).

Divisor-Sum Function \( \sigma(n) \)

The divisor-sum function \(\sigma(n)\) gives the sum of all positive divisors of \(n\).

Proper Divisors

Proper divisors are all positive divisors except the number itself.

Aliquot Sum

The aliquot sum is the sum of the proper divisors:

\[ s(n)=\sigma(n)-n \]

Perfect Numbers

A perfect number is equal to the sum of its proper divisors. The first perfect number is 6.

Abundant Numbers

An abundant number has a proper-divisor sum greater than the number itself. The smallest abundant number is 12.

Deficient Numbers

A deficient number has a proper-divisor sum smaller than the number itself. Every prime number is deficient.

Frequently Asked Questions

What is the sum of divisors?

The sum of divisors is the sum of all positive divisors of a number, including 1 and the number itself. It is represented by \(\sigma(n)\).

How do you calculate the sum of divisors?

You can find all positive divisors and add them together. Alternatively, if the prime factorization is known, the divisor-sum formula can calculate \(\sigma(n)\) directly from the prime factors.

What is the formula for the sum of divisors?

If \(n=p_1^{a_1}p_2^{a_2}\cdots p_k^{a_k}\), then:

\[ \sigma(n) = \prod_{i=1}^{k} \left( 1+p_i+p_i^2+\cdots+p_i^{a_i} \right) \]

What are proper divisors?

Proper divisors are all positive divisors of a number except the number itself.

What is the aliquot sum?

The aliquot sum is the sum of the proper divisors. It is calculated as:

\[ s(n)=\sigma(n)-n \]

What is a perfect number?

A perfect number is a positive integer whose proper divisors add up exactly to the number itself. For example, 6 is perfect because \(1+2+3=6\).

What is an abundant number?

An abundant number has a sum of proper divisors greater than the number itself. For example, 12 is abundant because:

\[ 1+2+3+4+6=16>12 \]

What is a deficient number?

A deficient number has a sum of proper divisors smaller than the number itself.

What does \( \sigma(n) \) mean?

\(\sigma(n)\) is the standard notation for the divisor-sum function. It gives the sum of all positive divisors of \(n\).

What does \( \tau(n) \) mean?

\(\tau(n)\) is commonly used for the divisor-counting function. It gives the number of positive divisors of \(n\).

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