On -amicable pairs |
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Authors: | Graeme L. Cohen Herman J. J. te Riele. |
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Affiliation: | School of Mathematical Sciences, University of Technology, Sydney, PO Box 123, Broadway, NSW 2007, Australia ; CWI, Department of Modeling, Analysis and Simulation, Kruislaan 413, 1098 SJ Amsterdam, The Netherlands |
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Abstract: | Let denote Euler's totient function, i.e., the number of positive integers and prime to . We study pairs of positive integers with such that for some integer . We call these numbers -amicable pairs with multiplier , analogously to Carmichael's multiply amicable pairs for the -function (which sums all the divisors of ). We have computed all the -amicable pairs with larger member and found pairs for which the greatest common divisor is squarefree. With any such pair infinitely many other -amicable pairs can be associated. Among these pairs there are so-called primitive -amicable pairs. We present a table of the primitive -amicable pairs for which the larger member does not exceed . Next, -amicable pairs with a given prime structure are studied. It is proved that a relatively prime -amicable pair has at least twelve distinct prime factors and that, with the exception of the pair , if one member of a -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 -amicable pairs with larger member , the largest pair consisting of two 46-digit numbers. |
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Keywords: | Euler's totient function $phi $--amicable pairs |
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