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Math Help - Square free numbers

  1. #1
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    Square free numbers

    Can anyone help with this problem?
    a function is defined to be abundant
    Last edited by miz.perfect84; August 11th 2009 at 11:15 AM.
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  2. #2
    Senior Member TheAbstractionist's Avatar
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    Do you mean find F(p^{\color{red}r\color{black}})\,?
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  3. #3
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    yes

    yes, sorry, i did type that in word but i think when i copied and pasted it it didn't come out right, thanks for that
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  4. #4
    Senior Member TheAbstractionist's Avatar
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    The number of integers between 1 and p^r which are not coprime with p^r are p,\,p^2,\,p^3,\,\ldots,\,p^r. Of these, only p is such that \gcd(p,p^r)=p is square-free. So the integers 1\le a\le p^r such that \gcd(a,p^r) is not square-free are the r-1 integers p^2,\,p^3,\,\ldots,\,p^r.

    \therefore\ F(p^r)=p^r-(r-1)=p^r-r+1.
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