The Sum of Positive Divisors of an Integer Examples 1

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)
\begin{align} \quad \sigma(n) = \sum_{d \mid n}_{d > 0} d \end{align}

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)
\begin{align} \sigma (9440) & = \sigma (2^5) \sigma (5) \sigma (59) \\ \sigma (9440) & = \left ( \frac{2^6 - 1}{2 - 1} \right ) (6)(60) \\ \sigma (9440) & = (63)(6)(60) \\ \sigma (9440) & = 22680 \end{align}

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)
\begin{align} \sigma (55202) & = \sigma (2) \sigma (7) \sigma (3943) \\ \sigma (55202) & = (3)(8)(3944) \\ \sigma (55202) & = 94656 \end{align}

Example 3

Calculate $\sigma (111111)$.

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

(4)
\begin{align} \sigma (111111) & = \sigma (3) \sigma (7) \sigma (11) \sigma (13) \sigma (37) \\ \sigma (111111) & = (4)(8)(12)(14)(38) \\ \sigma (111111) & = 204288 \end{align}

Example 4

Calculate $\sigma (712327)$.

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

(5)
\begin{align} \sigma (712327) & = \sigma (7) \sigma (11^2) \sigma (29^2) \\ \sigma (712327) & = (8) \left ( \frac{11^3 - 1}{11 - 1} \right )\left ( \frac{29^3 - 1}{29 - 1} \right ) \\ \sigma (712327) & = (8) (133)(871) \\ \sigma (712327) & = 926744 \end{align}