Abstract
Let \(\mathcal{A}(x)\) denote the set of integers n ≤ x that belong to an amicable pair. We show that \(\#\mathcal{A}(x) \leq x/e^{\sqrt{\log x}}\) for all sufficiently large x.
To Professor Helmut Maier on his 60th birthday
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
A. Anavi, P. Pollack, C. Pomerance, On congruences of the form \(\sigma (n) \equiv a\pmod n\). IJNT 9, 115–124 (2012)
W.D. Banks, J.B. Friedlander, C. Pomerance, I.E. Shparlinski, Multiplicative structure of values of the Euler function, in High Primes and Misdemeanours: Lectures in Honour of the 60th Birthday of Hugh Cowie Williams, ed. by A. van der Poorten (American Mathematical Society, Providence, 2004). Fields Inst. Commun. 41, 29–47 (2004)
E.R. Canfield, P. Erdős, C. Pomerance, On a problem of Oppenheim concerning ‘Factorisatio Numerorum’. J. Number Theory 17, 1–28 (1983)
N.G. de Bruijn, On the number of integers ≤ x and free of prime factors > y. Nederl. Akad. Wetensch. Proc. Ser. A 54, 50–60 (1951)
P. Erdős, On amicable numbers. Publ. Math. Debr. 4, 108–111 (1955)
P. Erdős, C. Pomerance, A. Sárközy, On locally repeated values of certain arithmetic functions, II. Acta Math. Hungar. 49, 251–259 (1987)
P. Erdős, G.J. Rieger, Ein Nachtrag über befreundete Zahlen. J. Reine Angew. Math. 273, 220 (1975)
S.W. Graham, J.J. Holt, C. Pomerance, On the solutions to \(\varphi (n) =\varphi (n + k)\), in Number Theory in Progress, ed. by K. Gyory, H. Iwaniec, J. Urbanowicz, vol. 2 (de Gruyter, Berlin/New York, 1999), pp. 867–882
H.J. Kanold, Über die Dichte der Mengen der vollkommenen und der befreundeten Zahlen. Math. Z. 61, 180–185 (1954)
Y. Lamzouri, Smooth values of the iterates of the Euler phi-function. Can. J. Math. 59, 127–147 (2007)
P. Pollack, L. Thompson, Practical pretenders. Publ. Math. Debr. 82, 651–667 (2013)
P. Pollack, J. Vandehey, Some normal numbers given by arithmetic functions Canad. Math. Bull. 58, 160–173 (2015)
C. Pomerance, On composite n for which \(\varphi (n)\mid n - 1\), I. Acta Arith. 28, 387–389 (1976)
C. Pomerance, On the distribution of amicable numbers. J. Reine Angew. Math. 293/294, 217–222 (1977)
C. Pomerance, On the distribution of amicable numbers, II. J. Reine Angew. Math. 325, 183–188 (1981)
G.J. Rieger, Bemerkung zu einem Ergebnis von Erdős über befreundete Zahlen. J. Reine Angew. Math. 261, 157–163 (1973)
T. Yamada, On equations \(\sigma (n) =\sigma (n + k)\) and \(\varphi (n) =\varphi (n + k)\). arXiv:1001.2511
Acknowledgements
The author would like to thank Hanh Nguyen, Paul Pollack, and Lola Thompson for their interest in this work.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Pomerance, C. (2015). On Amicable Numbers. In: Pomerance, C., Rassias, M. (eds) Analytic Number Theory. Springer, Cham. https://doi.org/10.1007/978-3-319-22240-0_19
Download citation
DOI: https://doi.org/10.1007/978-3-319-22240-0_19
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-22239-4
Online ISBN: 978-3-319-22240-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)