# The Sum of Positive Divisors of an Integer Examples 1

Recall from The Sum of Positive Divisors of an Integer page that if $n \in \mathbb{Z}$ then $\sigma(n)$ denotes the sum of all positive divisors of $n$ and is explicitly defined by:

(1)We also looked at some very simple properties of this function:

- If $p$ is a prime number then $\sigma(p) = 1 + p$.

- If $p$ is a prime number then $\sigma(p^2) = 1 + p + p^2$.

- If $p$ is a prime number and $k \in \mathbb{N}$ then $\displaystyle{\sigma(p^k) = 1 + p + ... + p^k = \frac{p^{k+1} - 1}{p - 1}}$.

- If $p$ and $q$ are prime numbers then $\sigma(pq) = \sigma(p) \sigma(q)$.

We will now look at some examples of determining $\sigma(n)$ for various $n$.

## Example 1

**Calculate $\sigma (9440)$.**

We first find the prime power decomposition of $9440$ In this case $9440 = 2^5 \cdot 5 \cdot 59$. Hence:

(2)## Example 2

**Calculate $\sigma (55202)$.**

Once again, we find that the prime power decomposition of $55202$ is $55202 = 2 \cdot 7 \cdot 3943$. Hence:

(3)## Example 3

**Calculate $\sigma (111111)$.**

The prime power decomposition of $111111$ is $111111 = 3 \cdot 7 \cdot 11 \cdot 13 \cdot 37$. Hence:

(4)## Example 4

**Calculate $\sigma (712327)$.**

The prime power decomposition of $712327$ is $712327 = 7 \cdot 11^2 \cdot 29^2$. Hence:

(5)