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Question Number 174566 by RajiYusuf last updated on 04/Aug/22

Answered by aleks041103 last updated on 05/Aug/22

$$f\left(n\right)=2n-\sigma \left(n\right)$$$$\sigma \left(n\right)=\sum _{i>0,i\mid n}i$$$$\sigma \left(p^2\right)=1+p+p^2$$$$\Rightarrow f\left(p^2\right)=2p^2-(1+p+p^2)=p^2-p-1$$$$p^2-p-1=0$$$$\Rightarrow p=\frac{1\pm \sqrt{5}}{2}=-0.618...;1.618...$$$$\Rightarrow p⩾2\Rightarrow p^2-p-1>0$$$$butifpisprime,thenp⩾2.$$$$\Rightarrow f\left(p^2\right)=2\left(p^2\right)-\sigma \left(p^2\right)>0$$$$\Rightarrow \sigma \left(n\right)<2nforn=p^2,wherepisprime$$

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