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Mathematics of Computation

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Primitive $ \alpha $-abundant numbers


Author: Graeme L. Cohen
Journal: Math. Comp. 43 (1984), 263-270
MSC: Primary 11A25
DOI: https://doi.org/10.1090/S0025-5718-1984-0744936-X
MathSciNet review: 744936
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Abstract: A number N is primitive $ \alpha $-abundant if $ \sigma (M)/M < \alpha \leqslant \sigma (N)/N$ for all proper divisors M of N. In this paper, we tabulate, for $ 1 < \alpha \leqslant 5.4$, all such N for which $ \sigma (N)/N$ is greatest. We show that, if N is primitive $ \alpha $-abundant and $ \alpha > 1.6$, then $ \sigma (N)/N < \alpha + \min \{ \frac{2}{5},3\alpha /2{e^{5\alpha /9}}\} $.


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DOI: https://doi.org/10.1090/S0025-5718-1984-0744936-X
Article copyright: © Copyright 1984 American Mathematical Society