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Math Help - Divisor summatory function

  1. #1
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    Bogota D.D
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    Divisor summatory function

    Hi everybody.

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

    Thanks.
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  2. #2
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    Re: Divisor summatory function

    Hi Fernando,

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

    - Hollywood
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  3. #3
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    Bogota D.D
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    Re: Divisor summatory function

    Thanks, the proof is by contradiction, is it not?

    Thanks for your suggestion.
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  4. #4
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    Crna Gora
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    Re: Divisor summatory function

    Quote Originally Posted by Fernando View Post
    Hi everybody.

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

    Thanks.
    \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 .
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