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Advances in aliquot sequences

Author(s): Manuel Benito; Juan L. Varona.
Journal: Math. Comp. 68 (1999), 389-393.
MSC (1991): Primary 11Y55; Secondary 11A25
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Abstract: In this paper we describe some advances in the knowledge of the behavior of aliquot sequences starting with a number less than $10000$. For some starting values, it is shown for the first time that the sequence terminates. The current record for the maximum of a terminating sequence is located in the one starting at 4170; it converges to 1 after 869 iterations getting a maximum of 84 decimal digits at iteration 289.

References:

1.
E. Catalan, Propositions et questions diverses, Bull. Soc. Math. France 18 (1887-88), 128-129.

2.
L. E. Dickson, Theorems and tables on the sum of the divisors of a number, Quart. J. Math. 44 (1913), 264-296.

3.
P. Erd\H{o}s, On asymptotic properties of aliquot sequences, Math. Comp. 30 (1976), 641-645. MR 53:7919

4.
A. W. P. Guy and R. K. Guy, A record aliquot sequence, in Computation 1943-1993: A Half-Century of Computational Mathematics (Vancouver, 1993), Proc. Sympos. Appl. Math. 48 (1994), Amer. Math. Soc., Providence, RI, 1994, pp. 557-559. MR 95k:11164

5.
R. K. Guy, Unsolved Problems in Number Theory (2nd ed.), Springer-Verlag, 1994. MR 96e:11002

6.
R. K. Guy and J. L. Selfridge, What drives an aliquot sequence?, Math. Comp. 29 (1975), 101-107. Corrigendum, ibid. 34 (1980), 319-321. MR 52:5542; MR 81f:10008

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Additional Information:

Manuel Benito
Affiliation: Departamento de Matemáticas y Computación, Universidad de La Rioja, Edificio J. L. Vives, Calle Luis de Ulloa s/n, 26004 Logroño, Spain
Email: mbenito@dmc.unirioja.es

Juan L. Varona
Affiliation: Departamento de Matemáticas y Computación, Universidad de La Rioja, Edificio J. L. Vives, Calle Luis de Ulloa s/n, 26004 Logroño, Spain
Email: jvarona@dmc.unirioja.es

DOI: 10.1090/S0025-5718-99-00991-6
PII: S 0025-5718(99)00991-6
Keywords: Aliquot sequences, sum of divisors, perfect number, amicable pair, sociable numbers, aliquot cycles
Received by editor(s): October 25, 1996
Received by editor(s) in revised form: February 26, 1997 and July 24, 1997
Additional Notes: Research of the second author supported by grant PB93-0228-C02-02 of the DGICYT
Copyright of article: Copyright 1999, American Mathematical Society

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