Additional Examples of Calculating σ(n) for Large Integers

We will be applying the formulas $\sigma (p^n) = \frac{p^{n + 1} - 1}{p - 1}$ and the case that $\sigma (p^1) = p + 1$ frequently on this page. Be sure to know it inside and out!

## Example 1

Calculate $\sigma (9440)$.

We must always find the prime power decomposition of 9440. In this case, 9440 = 25 x 5 x 59. Hence:

(1)
\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 to be 55202 = 2 x 7 x 3943. Hence:

(2)
\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 = 3 x 7 x 11 x 13 x 37. Hence:

(3)
\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 = 7 x 112 x 292. Hence:

(4)
\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}