On weird and pseudoperfect numbers

Benkoski, S. J.; Erdős, P.

doi:10.1090/S0025-5718-1974-0347726-9

On weird and pseudoperfect numbers
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by S. J. Benkoski and P. Erdős PDF

Math. Comp. 28 (1974), 617-623 Request permission

Corrigendum: Math. Comp. 29 (1975), 673-674.
Corrigendum: Math. Comp. 29 (1975), 673.

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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