How to calculate $\sigma(x^k)$?

Question

If $n = 2^k$, then $\sigma(n) = 2 \times 2^k - 1$,

For example:

$$\sigma(28) = \sigma(2^2) \times \sigma(7) = 7 \times 8 = 56$$

But how about:

$$\sigma(100) = \sigma(2^2) \times \sigma(5^2) = 7 \times x$$

I couldn't figure out how to calculate $\sigma(x^k)$

Answer 1

Hint: $\sigma$ is multiplicative and $\sigma(p^k)=1+p+p^2+ \cdots +p^{k-1}+p^k=\frac{p^{k+1}-1}{p-1}$, where $p$ is prime.