Abstract
Let s(n) denote the sum of the proper divisors of n. Set s 0(n) = n, and for k > 0, put s k (n) := s(s k-1(n)) if s k-1(n) > 0. Thus, perfect numbers are those n with s(n) = n and amicable numbers are those n with s(n) ≠ n but s 2(n) = n. We prove that for each fixed k ≥ 1, the set of n which divide s k (n) has density zero, and similarly for the set of n for which s k (n) divides n. These results generalize the theorem of Erdős that for each fixed k, the set of n for which s k (n) = n has density zero.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Cohen G.L., Gretton S.F., Hagis P. Jr: Multiamicable numbers. Math. Comput. 64, 1743–1753 (1995)
Erdős P.: Some remarks about additive and multiplicative functions. Bull. Am. Math. Soc. 52, 527–537 (1946)
Erdős P.: On the distribution of numbers of the form σ (n)/n and on some related questions. Pac. J. Math. 52, 59–65 (1974)
Erdős P.: On asymptotic properties of aliquot sequences. Math. Comput. 30, 641–645 (1976)
Kanold H.-J.: Über die Dichten der Mengen der vollkommenen und der befreundeten Zahlen. Math. Z. 61, 180– (1954)
Kanold H.-J.: Über mehrfach vollkommene Zahlen. J. Reine Angew. Math. 194, 218–220 (1955)
Kobayashi M., Pollack P., Pomerance C.: On the distribution of sociable numbers. J. Number Theory 129, 1990–2009 (2009)
Pomerance C.: On the distribution of amicable numbers. J. Reine Angew. Math. 293/294, 217–222 (1977)
Wirsing E.: Bemerkung zu der Arbeit über vollkommene Zahlen. Math. Ann. 137, 316–318 (1959)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by U. Zannier.
The author is supported by NSF award DMS-0802970.
Rights and permissions
About this article
Cite this article
Pollack, P. On some friends of the sociable numbers. Monatsh Math 162, 321–327 (2011). https://doi.org/10.1007/s00605-010-0240-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00605-010-0240-x