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Math Help - Abundant Integer

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    Abundant Integer

    List all the abundant integers up to 200
    Last edited by miz.perfect84; August 11th 2009 at 11:20 AM.
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  2. #2
    Senior Member TheAbstractionist's Avatar
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    Quote Originally Posted by miz.perfect84 View Post
    An integer n with the property σ (n) > 2n is sain to be abundant.
    Suppose that n=\color{red}2^r\color{black}p, where r ≥ 1 and p is odd prime. Show that for every odd prime p we can find values of r for which n is abundant. List all the abundant numbers of this form which are less than 100.

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    For each p there always exist integers r such that 2^{r+1}>p+1. Hence there exist integers r such that

    \sigma(n)\ =\ \sigma(2^rp)
    =\ \sigma(2^r)\sigma(p)
    =\ (1+2+2^2+\cdots+2^r)(p+1)
    =\ (2^{r+1}-1)(p+1)
    =\ 2^{r+1}p+2^{r+1}-p-1
    >\ 2^{r+1}p
    =\ 2n
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