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Mathematics of Computation

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On asymptotic properties of aliquot sequences


Author: P. Erdős
Journal: Math. Comp. 30 (1976), 641-645
MSC: Primary 10A20
DOI: https://doi.org/10.1090/S0025-5718-1976-0404115-8
MathSciNet review: 0404115
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Abstract: Put $ {s^{(1)}}(n) = \sigma (n) - n,\sigma (n) = {\Sigma _{d/n}}d$. $ {s^k}(n) = {s^{(1)}}({s^{(k - 1)}}(n))$. In this note we prove that for every k the density of integers satisfying

$\displaystyle {s^k}(n) = (1 + \sigma (1))n{((\sigma (n) - n)/n)^k}$

is 1. Several unsolved problems are stated.

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DOI: https://doi.org/10.1090/S0025-5718-1976-0404115-8
Article copyright: © Copyright 1976 American Mathematical Society