How to Calculate Σ N in Number Theory

What is σ(n)?

The sum of divisors function, denoted σ(n), is defined as the sum of all positive divisors of the positive integer n. A divisor of n is an integer that divides n without leaving a remainder. For example, the divisors of 6 are 1, 2, 3, and 6.

There are two common variants of the sum of divisors function:

• σ₀(n): Sum of all positive divisors of n (including 1 and n itself).
• σ₁(n): Sum of all positive divisors of n excluding n itself (sometimes called the "aliquot sum").

In this guide, we'll focus on σ₀(n), which is simply referred to as σ(n).

How to Calculate σ(n)

Calculating σ(n) involves finding all the positive divisors of n and then summing them up. Here's a step-by-step method to compute σ(n):

1. Find all divisors of n: Start by listing all integers from 1 to n that divide n without leaving a remainder.
2. Sum the divisors: Add up all the divisors you've found in the previous step.

For small values of n, this method is straightforward. However, for larger numbers, finding all divisors can be time-consuming. In such cases, you can use the prime factorization method, which is more efficient.

Prime Factorization Method

If you know the prime factorization of n, you can use the following formula to calculate σ(n):

This formula works because the sum of divisors function is multiplicative, meaning that σ(ab) = σ(a)σ(b) when a and b are coprime.

Examples

Let's look at a few examples to understand how to calculate σ(n).

Example 1: n = 6

The positive divisors of 6 are 1, 2, 3, and 6. Therefore, σ(6) = 1 + 2 + 3 + 6 = 12.

Example 2: n = 12

The positive divisors of 12 are 1, 2, 3, 4, 6, and 12. Therefore, σ(12) = 1 + 2 + 3 + 4 + 6 + 12 = 28.

Example 3: n = 15

The positive divisors of 15 are 1, 3, 5, and 15. Therefore, σ(15) = 1 + 3 + 5 + 15 = 24.

Using the prime factorization method, we can verify these results:

• For n = 6 = 2 × 3, σ(6) = (1 + 2) × (1 + 3) = 3 × 4 = 12.
• For n = 12 = 2² × 3, σ(12) = (1 + 2 + 4) × (1 + 3) = 7 × 4 = 28.
• For n = 15 = 3 × 5, σ(15) = (1 + 3) × (1 + 5) = 4 × 6 = 24.

Applications

The sum of divisors function has several important applications in number theory and related fields:

• Perfect numbers: A number n is called perfect if σ(n) = 2n. The smallest perfect number is 6, since σ(6) = 12 = 2 × 6.
• Abundant and deficient numbers: A number n is called abundant if σ(n) > 2n and deficient if σ(n) < 2n.
• Prime numbers: For a prime number p, σ(p) = 1 + p, since the only divisors are 1 and p itself.
• Number theory research: The sum of divisors function is used in various proofs and theorems in number theory.