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 $\sigma(x^k)$

You appear to know that $\sigma$ is multiplicative so I guess the question is how do you find $\sigma(p^k)$ for a prime $p$. To answer this I need to ask how you found $\sigma(2^k)$. — fretty, Dec 09 '14 at 19:50
$\sigma(2^k) = 2 \times 2^k-1$. For $k = 2$, $2 \times 2^2-1 = 8-1 = 7$. But I couldn't figure this out for other bases. — Patrick Bard, Dec 09 '14 at 19:54
But I couldn't actually understand this formula: $\sigma_x(n) = \prod_{i=1}^{r} \frac{p_{i}^{(a_{i}+1)x}-1}{p_{i}^x-1}$. Now I see... — Patrick Bard, Dec 09 '14 at 20:09
It is proved the exact same way it is proved for $2$ as the base...this is why I couldn't understand your issue so asked how you somehow managed to do that one particular case without seeing how to do the rest. — fretty, Dec 09 '14 at 20:10
But if you are just referencing the formulae from wiki then it makes more sense! — fretty, Dec 09 '14 at 20:11

score 2 · Answer 1 · answered Dec 09 '14 at 19:57

2

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.

answered Dec 09 '14 at 19:57

Nicky Hekster

I knew that for the case $\sigma(5^2) = 31$ as $7 \times 31 = 217$ which is the sum of divisors of 100. But as I did not know how to get 31 I omitted this number from the question. Thank you very much. – Patrick Bard Dec 09 '14 at 20:03
This is what I was trying to get at...the proof is no different at all for other bases. – fretty Dec 09 '14 at 20:08
Patrick, you are welcome! – Nicky Hekster Dec 09 '14 at 20:08
@fretty Yes, I know. But I only knew the base 2. I am sorry, but I couldn't understand the formula I commented above on my first look, I only understood the base 2 example. – Patrick Bard Dec 09 '14 at 20:10

1 Answers1