# Thread: σ sum of divisors proof

1. ## σ sum of divisors proof

σ is an odd integer if and only if n is a perfecdt square or twice a perfect square.

[Hint: If p i s an odd prime, then 1+p+p^2+...+p^k is odd only when k is even]

2. Could you please explain the link between σ, $\displaystyle n$ and $\displaystyle p$ ? I take that σ is the sum of the divisors of $\displaystyle n$, is that right ?

3. Originally Posted by MichaelG
σ is an odd integer if and only if n is a perfecdt square or twice a perfect square.

[Hint: If p i s an odd prime, then 1+p+p^2+...+p^k is odd only when k is even]

It is a basic fact that if $\displaystyle n=p_1^{a_1}\cdot \ldots \cdot p_k^{a_k}\in\mathbb{N}$ is the prime decomposition of a natural number, then $\displaystyle \sigma(n)=\prod\limits_{i=1}^k\frac{p_i^{a_i+1}-1}{p_i-1}=\prod\limits_{i=1}^k\left(\sum\limits_{m=0}^{a_ i}p_i^m\right)$

Well, now just use the hint...

Tonio

,

,

,

,

### Prove that σ(n) is an odd integer if and only if n is a perfect square or twice a perfect square.

Click on a term to search for related topics.