Journal of Integer Sequences, Vol. 16 (2013), Article 13.8.8

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


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 22k+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} \,

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(Concerned with sequences A062699 A068390 A104901.)

Received June 13 2013; revised versions received August 4 2013; September 19 2013; September 26 2013. Published in Journal of Integer Sequences, October 13 2013.

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