# Divisor summatory function

• Apr 10th 2012, 02:46 PM
Fernando
Divisor summatory function
Hi everybody.

Show that for some positive integer $\displaystyle n$ is composite iff $\displaystyle \sigma(n)>n+\sqrt{n}$.

Thanks.
• Apr 10th 2012, 08:53 PM
hollywood
Re: Divisor summatory function
Hi Fernando,

Try thinking about what $\displaystyle \sigma(n)$ is when $\displaystyle n$ is not composite (i.e. it is either prime or 1).

- Hollywood
• Apr 11th 2012, 02:52 AM
Fernando
Re: Divisor summatory function
Thanks, the proof is by contradiction, is it not?

• Apr 11th 2012, 03:18 AM
princeps
Re: Divisor summatory function
Quote:

Originally Posted by Fernando
Hi everybody.

Show that for some positive integer $\displaystyle n$ is composite iff $\displaystyle \sigma(n)>n+\sqrt{n}$.

Thanks.

$\displaystyle \sigma(n) =\begin{cases}n+1, & \text{if }n\text{ is prime} \\\displaystyle\prod_{i=1}^r \left(1+p_i+p^2_i+\ldots p^{a_i}_i\right), & \text{if }n\text{ is composite}\end{cases}$

where r is the number of distinct prime factors of n .