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On Weird and Pseudoperfect Numbers

S. J. Benkoski and P. Erdös
Mathematics of Computation
Vol. 28, No. 126 (Apr., 1974), pp. 617-623
DOI: 10.2307/2005938
Stable URL: http://www.jstor.org/stable/2005938
Page Count: 7
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On Weird and Pseudoperfect Numbers
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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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