#### Vol. 6, No. 8, 2012

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On common values of $\phi(n)$ and $\sigma(m)$, II

### Kevin Ford and Paul Pollack

Vol. 6 (2012), No. 8, 1669–1696
##### Abstract

For each positive-integer valued arithmetic function $f$, let ${\mathsc{V}}_{f}\subset ℕ$ denote the image of $f$, and put ${\mathsc{V}}_{f}\left(x\right):={\mathsc{V}}_{f}\cap \left[1,x\right]$ and ${\mathsc{V}}_{f}\left(x\right):=#{\mathsc{V}}_{f}\left(x\right)$. Recently Ford, Luca, and Pomerance showed that ${\mathsc{V}}_{\varphi }\cap {\mathsc{V}}_{\sigma }$ is infinite, where $\varphi$ denotes Euler’s totient function and $\sigma$ is the usual sum-of-divisors function. Work of Ford shows that ${V}_{\varphi }\left(x\right)\asymp {V}_{\sigma }\left(x\right)$ as $x\to \infty$. Here we prove a result complementary to that of Ford et al. by showing that most $\varphi$-values are not $\sigma$-values, and vice versa. More precisely, we prove that, as $x\to \infty$,

$#\left\{n\le x:n\in {\mathsc{V}}_{\varphi }\cap {\mathsc{V}}_{\sigma }\right\}\le \frac{{V}_{\varphi }\left(x\right)+{V}_{\sigma }\left(x\right)}{{\left(loglogx\right)}^{1∕2+o\left(1\right)}}.$

##### Keywords
Euler function, totient, sum of divisors
##### Mathematical Subject Classification 2010
Primary: 11N37
Secondary: 11N64, 11A25, 11N36