Skip to main content Accessibility help
Hostname: page-component-59f8fd8595-tmn4r Total loading time: 0 Render date: 2023-03-22T09:16:08.135Z Has data issue: true Feature Flags: { "useRatesEcommerce": false } hasContentIssue true

7 - The First Function and Its Iterates

Published online by Cambridge University Press:  25 May 2018

Carl Pomerance
Mathematics Department, Dartmouth College, Hanover, NH 03755, USA
Steve Butler
Iowa State University
Joshua Cooper
University of South Carolina
Glenn Hurlbert
Virginia Commonwealth University
Get access


Image of the first page of this content. For PDF version, please use the ‘Save PDF’ preceeding this image.'
Connections in Discrete Mathematics
A Celebration of the Work of Ron Graham
, pp. 125 - 138
Publisher: Cambridge University Press
Print publication year: 2018

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)


1. A. R., Booker. Finite connected components of the aliquot graph. Math. Comp., to appear, DOI:
2. W., Bosma. Aliquot sequences with small starting values. arXiv:1604.03004 [math.NT].
3. W., Bosma and B., Kane. The aliquot constant. Quart J. Math. 63 (2012), 309–323.Google Scholar
4. E., Catalan. Propositions et questions diverses. Bull. Soc. Math. France 16 (1888), 128–129.Google Scholar
5. L. E., Dickson. Theorems and tables on the sum of the divisors of a number. Quart J. Math. 44 (1913), 264–296.Google Scholar
6. N. G. de, Bruijn. On the number of positive integers ≤ x and free of prime factors > y. Nederl. Acad. Wetensch. Proc. Ser. A 54 (1951), 50–60.Google Scholar
7. P., Erdős. Uber die Zahlen der Form σ Elem.Math. 28 (1973), 83–86.Google Scholar
8. P., Erdős. On asymptotic properties of aliquot sequences. Math. Comp. 30 (1976), 641–645.Google Scholar
9. 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, Birkhauser Boston, Boston, MA, 1990, pp. 165–204.Google Scholar
10. P., Erdős, F., Luca, and C., Pomerance. On the proportion of numbers coprime to a given integer. Anatomy of Integers. CRMProc. Lecture Notes, Vol. 46. Amer.Math. Soc., Providence, RI, 2008, pp. 47–64.Google Scholar
11. R. K., Guy and J. L., Selfridge. What drives an aliquot sequence? Math. Comp. 29 (1975), 101–107.Google Scholar
12. H., Halberstam and H.-E., Richert. Sieve Methods. Academic Press, London, 1974.Google Scholar
13. M., Kobayashi. On the Density of the Abundant Numbers. PhD dissertation, Dartmouth College, 2010.
14. M., Kobayashi, P., Pollack, and C., Pomerance. On the distribution of sociable numbers. J. Number Theory 129 (2009), 1990–2009.Google Scholar
15. H.W., Lenstra, Jr. Problem 6064. Amer. Math. Monthly 82 (1975), 1016. Solution by the proposer, op. cit. 84 (1977), 580.Google Scholar
16. F., Luca and C., Pomerance. Irreducible radical extensions and Euler function chains. Combinatorial Number Theory. de Gruyter, Berlin, 2007, pp. 351–361; Integers 7 (2007), no. 2, #A25.
17. F., Luca and C., Pomerance. The range of the sum-of-proper-divisors function. Acta Arith. 168 (2015), 187–199.Google Scholar
18. F., Luca and C., Pomerance. Local behavior of the composition of the aliquot and cototient functions. In Number Theory: In Honor of Krishna Alladi's 60th Birthday. G., Andrews and F., Garvan, eds. Spinger Science + Business Media, New York, forthcoming.
19. H. L., Montgomery and R. C., Vaugan. The exceptional set in Goldbach's problem, Acta Arith. 27 (1975), 353–370.Google Scholar
20. P., Pollack. On the greatest common divisor of a number and its sum of divisors. Michigan Math. J. 60 (2011), 199–214.Google Scholar
21. P., Pollack. Some arithmetic properties of the sum of proper divisors and the sum of prime divisors. Illinois J. Math. 58 (2014), 125–147.Google Scholar
22. P., Pollack and C., Pomerance. Some problems of Erdős on the sum-of-divisors function. Trans. Amer. Math. Soc. Ser. B 3 (2016), 1–26.Google Scholar
23. C., Pomerance. On the distribution of amicable numbers. J. Reine Angew. Math. 293/294 (1977), 217–222.Google Scholar
24. C., Pomerance. Primality testing:Variations on a theme of Lucas. Proceedings of the 13th Meeting of the Fibonacci Association, Congressus Numerantium 201 (2010), 301–312.Google Scholar
25. C., Pomerance. On amicable numbers. In Analytic Number Theory: In Honor of Helmut Maier's 60th birthday, M., Rassias and C., Pomerance, eds., Springer, Cham, Switzerland, 2015, pp. 321–327.Google Scholar
26. C., Pomerance and H.-S., Yang. Variant of a theorem of Erdős on the sum-of-properdivisors function. Math. Comp. 83 (2014), no. 288, 1903–1913.Google Scholar
27. L., Troupe. On the number of prime factors of values of the sum-of-proper-divisors function. J. Number Theory, 150 (2015), 120–135.Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the or variations. ‘’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats