On the Ratio of the Sum of Divisors and Euler’s Totient Function I

We prove that the only solutions to the equation $\sigma(n)=2\cdot\phi(n)$ with at most three distinct prime factors are 3, 35 and 1045. Moreover, there exist at most a finite number of solutions to $\sigma(n)=2\cdot\phi(n)$ with $\Omega(n)\le k$ , and there are at most 2^{2^k+k}-k squarefree solutions to $\phi(n) \big\vert \sigma(n)$ if $\omega(n)=k$ . Lastly the number of solutions to $\phi(n) \big\vert \sigma(n)$ as $x\rightarrow\infty$ is $O\left(x\exp\left(-\frac{1}{2}\sqrt{\log x} \, \right)\right)$ .

On the Ratio of the Sum of Divisors and Euler’s Totient Function I

Kevin A. Broughan and Daniel Delbourgo Department of Mathematics University of Waikato Private Bag 3105, Hamilton New Zealand

Kevin A. Broughan and Daniel Delbourgo
Department of Mathematics
University of Waikato
Private Bag 3105, Hamilton
New Zealand