The reciprocal sum of the amicable numbers
HTML articles powered by AMS MathViewer
- by Hanh My Nguyen and Carl Pomerance HTML | PDF
- Math. Comp. 88 (2019), 1503-1526 Request permission
Abstract:
In this paper, we improve on several earlier attempts to show that the reciprocal sum of the amicable numbers is small, showing this sum is $<215$.References
- Jonathan Bayless and Paul Kinlaw, Consecutive coincidences of Euler’s function, Int. J. Number Theory 12 (2016), no. 4, 1011–1026. MR 3484296, DOI 10.1142/S1793042116500639
- Jonathan Bayless and Dominic Klyve, On the sum of reciprocals of amicable numbers, Integers 11 (2011), no. 3, 315–332. MR 2988065, DOI 10.1515/integ.2011.025
- Olivier Bordellès, An inequality for the class number, JIPAM. J. Inequal. Pure Appl. Math. 7 (2006), no. 3, Article 87, 8. MR 2257286
- Jan Büthe, Estimating $\pi (x)$ and related functions under partial RH assumptions, Math. Comp. 85 (2016), no. 301, 2483–2498. MR 3511289, DOI 10.1090/mcom/3060
- J. Büthe, An analytic method for bounding $\psi (x)$, Math. Comp., to appear, https://doi.org/10.1090/mcom/3264. Also see arxiv.org 1511.02032[math.NT].
- S. Chernykh, Amicable pairs list, https://sech.me/ap/index.html.
- Pierre Dusart, Explicit estimates of some functions over primes, Ramanujan J. 45 (2018), no. 1, 227–251. MR 3745073, DOI 10.1007/s11139-016-9839-4
- P. Erdös, On amicable numbers, Publ. Math. Debrecen 4 (1955), 108–111. MR 69198
- H. M. Nguyen, The reciprocal sum of the amicable numbers, Senior honors thesis, Mathematics, Dartmouth College, 2014.
- Su Hee Kim and Carl Pomerance, The probability that a random probable prime is composite, Math. Comp. 53 (1989), no. 188, 721–741. MR 982368, DOI 10.1090/S0025-5718-1989-0982368-4
- Elvin Lee, On divisibility by nine of the sums of even amicable pairs, Math. Comp. 23 (1969), 545–548. MR 248074, DOI 10.1090/S0025-5718-1969-0248074-6
- J. D. Lichtman, The reciprocal sum of primitive nondeficient numbers, arXiv:1801.01925v2[math.NT], 2018.
- Jared D. Lichtman and Carl Pomerance, Explicit estimates for the distribution of numbers free of large prime factors, J. Number Theory 183 (2018), 1–23. MR 3715225, DOI 10.1016/j.jnt.2017.08.039
- H. L. Montgomery and R. C. Vaughan, The large sieve, Mathematika 20 (1973), 119–134. MR 374060, DOI 10.1112/S0025579300004708
- D. Platt and T. Trudgian, Improved bounds on Brun’s constant, arXiv:1803.01925v1 [math.NT].
- Carl Pomerance, On the distribution of amicable numbers. II, J. Reine Angew. Math. 325 (1981), 183–188. MR 618552, DOI 10.1515/crll.1981.325.183
- Carl Pomerance, On amicable numbers, Analytic number theory, Springer, Cham, 2015, pp. 321–327. MR 3467405
- J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64–94. MR 137689
Additional Information
- Hanh My Nguyen
- Affiliation: Applied Predictive Technologies, Inc., 4250 N Fairfax Dr., 11th Floor, Arlington, Virginia 22203
- Email: hanhmn91@gmail.com
- Carl Pomerance
- Affiliation: Mathematics Department, Dartmouth College, Hanover, New Hampshrie 03755
- MR Author ID: 140915
- Email: carl.pomerance@dartmouth.edu
- Received by editor(s): August 12, 2017
- Received by editor(s) in revised form: August 14, 2017, and January 30, 2018
- Published electronically: April 10, 2018
- © Copyright 2018 American Mathematical Society
- Journal: Math. Comp. 88 (2019), 1503-1526
- MSC (2010): Primary 11A25, 11N25, 11N64
- DOI: https://doi.org/10.1090/mcom/3362
- MathSciNet review: 3904154