Advances in aliquot sequences

Authors:
Manuel Benito and Juan L. Varona

Journal:
Math. Comp. **68** (1999), 389-393

MSC (1991):
Primary 11Y55; Secondary 11A25

DOI:
https://doi.org/10.1090/S0025-5718-99-00991-6

MathSciNet review:
1489967

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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we describe some advances in the knowledge of the behavior of aliquot sequences starting with a number less than . 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.

**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:
https://doi.org/10.1090/S0025-5718-99-00991-6

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

Article copyright:
© Copyright 1999
American Mathematical Society