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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

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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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1974-0347726-9
PII: S 0025-5718(1974)0347726-9
Keywords: Weird numbers, pseudoperfect numbers, primitive abundant numbers
Article copyright: © Copyright 1974 American Mathematical Society