Square free numbers

**miz.perfect84** · June 13th 2009, 12:37 PM

Can anyone help with this problem?
a function is defined to be abundant

**TheAbstractionist** · June 13th 2009, 05:08 PM

Do you mean find $F(p^{\color{red}r\color{black}})\,?$

**miz.perfect84** · June 14th 2009, 01:00 AM

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

**TheAbstractionist** · June 14th 2009, 02:30 AM

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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