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**Drexel28** What about this? Let $\displaystyle n\in\mathbb{N}$ be arbitrary. Then, $\displaystyle n=p_1^{\alpha_1}\cdots p_m^{\alpha_m}$ and so $\displaystyle \sigma(n)=\prod_{j=1}^{m}\frac{p_m^{\alpha_m+1}-1}{p_m-1}$. So, if $\displaystyle \sigma(n)=17$ it follows that $\displaystyle 17=\frac{p_m^{\alpha_m+1}-1}{p-1}$ from it's factorization properties. But, this means that $\displaystyle 17=1+\cdots+p_m^{\alpha_m}$ and thus $\displaystyle 16=p_m^1+\cdots+p_m^{\alpha_m}$ but this means that $\displaystyle 16=p_m(1+\cdots+p_m^{\alpha_m})$ and thus $\displaystyle p_m=2\implies 8=1+\cdots+p_m^{\alpha_m-1}$ which is ridicuoulous why?