Advances in aliquot sequences

Authors:
Manuel Benito and Juan L. Varona

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

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

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. Erdős,*On asymptotic properties of aliquot sequences*, Math. Comp.**30**(1976), no. 135, 641–645. MR**0404115**, 10.1090/S0025-5718-1976-0404115-8**4.**Andrew W. P. Guy and Richard K. Guy,*A record aliquot sequence*, Mathematics of Computation 1943–1993: a half-century of computational mathematics (Vancouver, BC, 1993) Proc. Sympos. Appl. Math., vol. 48, Amer. Math. Soc., Providence, RI, 1994, pp. 557–559. MR**1314890**, 10.1090/psapm/048/1314890**5.**Richard K. Guy,*Unsolved problems in number theory*, 2nd ed., Problem Books in Mathematics, Springer-Verlag, New York, 1994. Unsolved Problems in Intuitive Mathematics, I. MR**1299330****6.**Richard K. Guy and J. L. Selfridge,*What drives an aliquot sequence?*, Math. Comput.**29**(1975), 101–107. Collection of articles dedicated to Derrick Henry Lehmer on the occasion of his seventieth birthday. MR**0384669**, 10.1090/S0025-5718-1975-0384669-X

Richard K. Guy and J. L. Selfridge,*Corrigendum to: “What drives an aliquot sequence?” [Math. Comp. 29 (1975), 101–107; MR 52 #5542]*, Math. Comp.**34**(1980), no. 149, 319–321. MR**551309**, 10.1090/S0025-5718-1980-0551309-8

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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