Finite connected components of the aliquot graph

Booker, Andrew

doi:10.1090/mcom/3299

Finite connected components of the aliquot graph
HTML articles powered by AMS MathViewer

by Andrew R. Booker PDF

Math. Comp. 87 (2018), 2891-2902 Request permission

Abstract:

Conditional on a strong form of the Goldbach conjecture, we determine all finite connected components of the aliquot graph containing a number less than $10^9$, as well as those containing an amicable pair below $10^{14}$ or one of the known perfect or sociable cycles below $10^{17}$. Along the way we develop a fast algorithm for computing the inverse image of an even number under the sum-of-proper-divisors function.

References

D. J. Bernstein, How to find small factors of integers, 2002, http://cr.yp.to/papers.html#sf.
A. R. Booker, Web page with extended data, http://people.maths.bris.ac.uk/~maarb/aliquot/.
S. Chernykh, Amicable pairs list, http://sech.me/ap/.
J.-P. Delahaye, Nombres amiables et suites aliquotes, Pour la Science (2002), no. 292, 98–103.
P. Erdős, A. Granville, C. Pomerance, and C. Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory (Allerton Park, IL, 1989) Progr. Math., vol. 85, Birkhäuser Boston, Boston, MA, 1990, pp. 165–204. MR 1084181, DOI 10.1007/978-1-4612-3464-7_{1}3
Jan Feitsma, Pseudoprimes, http://www.janfeitsma.nl/math/psp2/index.
Jean-Luc Garambois, Les suites aliquotes, http://www.aliquotes.com/.
Wen Chao Lu, Exceptional set of Goldbach number, J. Number Theory 130 (2010), no. 10, 2359–2392. MR 2660899, DOI 10.1016/j.jnt.2010.03.017
David Moews, A list of aliquot cycles of length greater than $2$, http://djm.cc/sociable.txt.
Tomás Oliveira e Silva, Siegfried Herzog, and Silvio Pardi, Empirical verification of the even Goldbach conjecture and computation of prime gaps up to $4\cdot 10^{18}$, Math. Comp. 83 (2014), no. 288, 2033–2060. MR 3194140, DOI 10.1090/S0025-5718-2013-02787-1
Paul Pollack and Carl Pomerance, Some problems of Erdős on the sum-of-divisors function, Trans. Amer. Math. Soc. Ser. B 3 (2016), 1–26. MR 3481968, DOI 10.1090/btran/10
C. Pomerance, The first function and its iterates, Connections in Discrete Mathematics: A Celebration of the Work of Ron Graham (S. Butler, J. Cooper, and G. Hurlbert, eds.), Cambridge University Press, 2017, https://math.dartmouth.edu/~carlp/aliquot8.pdf.
J. Sesiano, Two problems of number theory in Islamic times, Arch. Hist. Exact Sci. 41 (1991), no. 3, 235–238. MR 1107382, DOI 10.1007/BF00348408
The PARI Group, Bordeaux, PARI/GP version 2.8.0, 2016, available from http://pari.math.u-bordeaux.fr/.