Journal of Integer Sequences, Vol. 14 (2011), Article 11.5.2

Quasi-Amicable Numbers are Rare

Paul Pollack
University of Illinois at Urbana-Champaign
Department of Mathematics
1409 West Green Street
Urbana, IL 61801


Define a quasi-amicable pair as a pair of distinct natural numbers each of which is the sum of the nontrivial divisors of the other, e.g., {48, 75}. Here nontrivial excludes both 1 and the number itself. Quasi-amicable pairs have been studied (primarily empirically) by Garcia, Beck and Najar, Lal and Forbes, and Hagis and Lord. We prove that the set of n belonging to a quasi-amicable pair has asymptotic density zero.

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(Concerned with sequences A000396 A005276 A007992 A015630 A063990 A126160.)

Received November 26 2010. revised version received April 16 2011. Published in Journal of Integer Sequences, April 21 2011.

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