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Multiamicable numbers


Authors: Graeme L. Cohen, Stephen F. Gretton and Peter Hagis
Journal: Math. Comp. 64 (1995), 1743-1753
MSC: Primary 11A25
DOI: https://doi.org/10.1090/S0025-5718-1995-1308449-6
MathSciNet review: 1308449
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Abstract: Multiamicable numbers are a natural generalization of amicable numbers: two numbers form a multiamicable pair if the sum of the proper divisors of each is a multiple of the other. Many other generalizations have been considered in the past. This paper reviews those earlier generalizations and gives examples and properties of multiamicable pairs. It includes a proof that the set of all multiamicable numbers has density 0.


References [Enhancements On Off] (What's this?)

  • [1] S. Battiato and W. Borho, Are there odd amicable numbers not divisible by three?, Math. Comp. 50 (1988), 633-637. MR 929559 (89c:11015)
  • [2] R. D. Carmichael, Review of History of the theory of numbers, Amer. Math. Monthly 26 (1919), 396-403. MR 1519382
  • [3] Graeme L. Cohen, On an integer's infinitary divisors, Math. Comp. 54 (1990), 395-411. MR 993927 (90e:11011)
  • [4] -, Generalisations of amicable numbers, Internal Report No. 36, School of Mathematical Sciences, University of Technology, Sydney, April 1992.
  • [5] Henri Cohen, On amicable and sociable numbers, Math. Comp. 24 (1970), 423-429. MR 0271004 (42:5887)
  • [6] L. E. Dickson, Amicable number triples, Amer. Math. Monthly 20 (1913), 84-91. MR 1517797
  • [7] Leonard Eugene Dickson, History of the theory of numbers, Vol. 1, Chelsea, New York, 1971.
  • [8] P. Erdős, On amicable numbers, Publ. Math. Debrecen 4 (1955), 108-111. MR 0069198 (16:998h)
  • [9] Achim Flammenkamp, New sociable numbers, Math. Comp. 56 (1991), 871-873. MR 1052094 (92a:11004)
  • [10] Richard K. Guy, Unsolved problems in number theory, Springer-Verlag, New York, 1981. MR 656313 (83k:10002)
  • [11] Peter Hagis, Jr, Unitary amicable numbers, Math. Comp. 25 (1971), 915-918. MR 0299551 (45:8599)
  • [12] C. Krishnamurthy, Some sets of amicable numbers of higher order, Indian J. Pure Appl. Math. 11(12) (1980), 1549-1553. MR 617829 (82m:10010)
  • [13] A. Makowski, On some equations involving functions $ \phi (n)$ and $ \sigma (n)$, Amer. Math. Monthly 67 (1960), 668-670; correction, ibid. 68 (1961), 650. MR 0130209 (24:A76)
  • [14] Thomas E. Mason, On amicable numbers and their generalizations, Amer. Math. Monthly 28 (1921), 195-200. MR 1519771
  • [15] Wayne L. McDaniel, On odd multiply perfect numbers, Boll. Un. Mat. Ital. 2 (1970), 185-190. MR 0262154 (41:6764)
  • [16] David Moews and Paul C. Moews, A search for aliquot cycles below $ {10^{10}}$, Math. Comp. 57 (1991), 849-855. MR 1094955 (92e:11151)
  • [17] Paul Poulet, La chasse aux nombres, Bruxelles, 1929.
  • [18] Herman J. J. te Riele, On generating new amicable pairs from given amicable pairs, Math. Comp. 42 (1984), 219-223. MR 725997 (85d:11107)
  • [19] W. Sierpiñski, Elementary theory of numbers, North-Holland, Amsterdam, 1988.
  • [20] Benjamin Franklin Yanney, Another definition of amicable numbers and some of their relations to Dickson's amicables, Amer. Math. Monthly 30 (1923), 311-315. MR 1520272

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1995-1308449-6
Article copyright: © Copyright 1995 American Mathematical Society

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