Can anyone help with this problem?

a function is defined to be abundant

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- Jun 13th 2009, 12:37 PMmiz.perfect84Square free numbers
Can anyone help with this problem?

a function is defined to be abundant - Jun 13th 2009, 05:08 PMTheAbstractionist
Do you mean find $\displaystyle F(p^{\color{red}r\color{black}})\,?$

- Jun 14th 2009, 01:00 AMmiz.perfect84yes
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

- Jun 14th 2009, 02:30 AMTheAbstractionist
The number of integers between 1 and $\displaystyle p^r$ which are not coprime with $\displaystyle p^r$ are $\displaystyle p,\,p^2,\,p^3,\,\ldots,\,p^r.$ Of these, only $\displaystyle p$ is such that $\displaystyle \gcd(p,p^r)=p$ is square-free. So the integers $\displaystyle 1\le a\le p^r$ such that $\displaystyle \gcd(a,p^r)$ is not square-free are the $\displaystyle r-1$ integers $\displaystyle p^2,\,p^3,\,\ldots,\,p^r.$

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