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arXiv · 1910.08452

On two conjectures of Sierpi\'nski concerning the arithmetic functions $\sigma$ and $\phi$

Kevin Ford·Sergei Konyagin
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Abstract

Let $\sigma(n)$ denote the sum of the positive divisors of $n$. We prove that for any positive integer $k$, there is a number $m$ for which the equation $\sigma(x)=m$ has exactly $k$ solutions, settling a conjecture of Sierpi\'nski from 1955. Additionally, it is shown that for every positive even $k$, there is a number $m$ for which the equation $\phi(x)=m$ has exactly $k$ solutions, where $\phi$ is Euler's function, making progress toward another conjecture of Sierpi\'nski from 1955.

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Kevin Ford, Sergei Konyagin. 2019-10-18. On two conjectures of Sierpi\'nski concerning the arithmetic functions $\sigma$ and $\phi$. https://arxiv.org/abs/1910.08452

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