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Mathematics of Computation

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On $\phi$-amicable pairs


Authors: Graeme L. Cohen and Herman J. J. te Riele
Journal: Math. Comp. 67 (1998), 399-411
MSC (1991): Primary 11A25, 11Y70
DOI: https://doi.org/10.1090/S0025-5718-98-00933-8
MathSciNet review: 1458219
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Abstract: Let $\phi (n)$ denote Euler’s totient function, i.e., the number of positive integers $<n$ and prime to $n$. We study pairs of positive integers $(a_{0},a_{1})$ with $a_{0}\le a_{1}$ such that $\phi (a_{0})=\phi (a_{1})=(a_{0}+a_{1})/k$ for some integer $k\ge 1$. We call these numbers $\phi$–amicable pairs with multiplier $k$, analogously to Carmichael’s multiply amicable pairs for the $\sigma$–function (which sums all the divisors of $n$). We have computed all the $\phi$–amicable pairs with larger member $\le 10^{9}$ and found $812$ pairs for which the greatest common divisor is squarefree. With any such pair infinitely many other $\phi$–amicable pairs can be associated. Among these $812$ pairs there are $499$ so-called primitive $\phi$–amicable pairs. We present a table of the $58$ primitive $\phi$–amicable pairs for which the larger member does not exceed $10^{6}$. Next, $\phi$–amicable pairs with a given prime structure are studied. It is proved that a relatively prime $\phi$–amicable pair has at least twelve distinct prime factors and that, with the exception of the pair $(4,6)$, if one member of a $\phi$–amicable pair has two distinct prime factors, then the other has at least four distinct prime factors. Finally, analogies with construction methods for the classical amicable numbers are shown; application of these methods yields another 79 primitive $\phi$–amicable pairs with larger member $>10^{9}$, the largest pair consisting of two 46-digit numbers.


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Additional Information

Graeme L. Cohen
Affiliation: School of Mathematical Sciences, University of Technology, Sydney, PO Box 123, Broadway, NSW 2007, Australia
Email: glc@maths.uts.edu.au

Herman J. J. te Riele
Affiliation: CWI, Department of Modeling, Analysis and Simulation, Kruislaan 413, 1098 SJ Amsterdam, The Netherlands
Email: herman@cwi.nl

Keywords: Euler’s totient function, $\phi$–amicable pairs
Received by editor(s): November 28, 1995
Received by editor(s) in revised form: May 10, 1996
Article copyright: © Copyright 1998 American Mathematical Society