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Mathematics of Computation

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On weird and pseudoperfect numbers

Authors: S. J. Benkoski and P. Erdős
Journal: Math. Comp. 28 (1974), 617-623
MSC: Primary 10A40
Corrigendum: Math. Comp. 29 (1975), 673-674.
Corrigendum: Math. Comp. 29 (1975), 673.
MathSciNet review: 0347726
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Abstract: If n is a positive integer and $\sigma (n)$ denotes the sum of the divisors of n, then n is perfect if $\sigma (n) = 2n$, abundant if $\sigma (n) \geqq 2n$ and deficient if $\sigma (n) < 2n$. n is called pseudoperfect if n is the sum of distinct proper divisors of n. If n is abundant but not pseudoperfect, then n is called weird. The smallest weird number is 70. We prove that the density of weird numbers is positive and discuss several related problems and results. A list of all weird numbers not exceeding ${10^6}$ is given.

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Keywords: Weird numbers, pseudoperfect numbers, primitive abundant numbers
Article copyright: © Copyright 1974 American Mathematical Society