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On primitive abundant numbers

Published online by Cambridge University Press:  09 April 2009

Graeme L. Cohen
Graeme L. Cohen
Affiliation:
School of Mathematical Sciences The New South Wales Institute of TechnologyP. O. Box 123, Broadway, NSW 2007, Australia
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Abstract

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Let n be a natural with largest component sd. We prove that if xσ(n) = yn + z (x, y, z given positive integers), n is not primitive (y/x)-abundant and n/s is not (y/x)-perfect, then n < 4(z + ½)3/27y (if z ≥ 175). All solutions are tabled for the equation xσ(n) - yn when x = 1, y ≥ 2, 1 ≤ z ≤ 210, and n is not primitive y-abundant. We also prove that if n is primitive (y/x)-abundant, then s3d < (yn/2)2. A number of results are proved concerning the range of σ(n)/n when n is primitive αabundant, for any real number α > 1. For example, then α(n/n < α + min {½ 3 α e-5a/9 /2) and σ(n)/n < α /1.6α/log n. All primitive abundant numbers n with α(n)/ ≥ 2.05 are listed.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 34 , Issue 1 , February 1983 , pp. 123 - 137
Copyright
Copyright © Australian Mathematical Society 1983

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