Sigma Function Odd iff Argument is Square or Twice Square

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Theorem

Let $\sigma: \Z \to \Z$ be the sigma function.

Then $\sigma \left({n}\right)$ is odd if and only if $n$ is either square or twice a square.


Proof

Let $n$ be an integer such that $n \ge 2$.

Let the prime decomposition of $n$ be:

$n = p_1^{k_1} p_2^{k_2} \ldots p_r^{k_r}$

Then from Sigma Function of Integer we have that:

$\displaystyle \sigma \left({n}\right) = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$

That is:

$\displaystyle \sigma \left({n}\right) = \prod_{1 \mathop \le i \mathop \le r} \left({1 + p_i + p_i^2 + \ldots + p_i^{k_i}}\right)$


Let $\sigma \left({n}\right)$ be odd.

Then all factors of $\displaystyle \prod_{i \mathop = 1}^r \left({1 + p_i + p_i^2 + \ldots + p_i^{k_i}}\right)$ are odd (and of course $\ge 3$).


For $1 + p_i + p_i^2 + \ldots + p_i^{k_i}$ to be odd, one of two conditions must hold:

$p_i$ is even (so that all terms of $1 + p_i + p_i^2 + \ldots + p_i^{k_i}$ are even except the $1$);
$k_i$ is even (so that $1 + p_i + p_i^2 + \ldots + p_i^{k_i}$ has an odd number of odd terms).

In the first case, that means $p_i^{k_i}$ is a power of $2$.

In the second case, that means $p_i^{k_i}$ is a square.

The result follows.


The argument reverses.

$\blacksquare$