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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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On $\phi$-amicable pairs
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by Graeme L. Cohen and Herman J. J. te Riele PDF
Math. Comp. 67 (1998), 399-411 Request permission


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:
  • Herman J. J. te Riele
  • Affiliation: CWI, Department of Modeling, Analysis and Simulation, Kruislaan 413, 1098 SJ Amsterdam, The Netherlands
  • Email:
  • Received by editor(s): November 28, 1995
  • Received by editor(s) in revised form: May 10, 1996
  • © Copyright 1998 American Mathematical Society
  • Journal: Math. Comp. 67 (1998), 399-411
  • MSC (1991): Primary 11A25, 11Y70
  • DOI:
  • MathSciNet review: 1458219