#### 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
##### Milestones
Received: 29 November 2010
Revised: 30 November 2011
Accepted: 30 January 2012
Published: 14 December 2012
##### Authors
 Kevin Ford Department of Mathematics University of Illinois 1409 West Green Street Urbana, IL 61801 United States Paul Pollack Department of Mathematics University of Georgia Boyd Graduate Studies Research Center Athens, GA 30602 United States