On weird and pseudoperfect numbers

Authors:
S. J. Benkoski and P. Erdős

Journal:
Math. Comp. **28** (1974), 617-623

MSC:
Primary 10A40

DOI:
https://doi.org/10.1090/S0025-5718-1974-0347726-9

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

Corrigendum:
Math. Comp. **29** (1975), 673.

MathSciNet review:
0347726

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Abstract | References | Similar Articles | Additional Information

Abstract: If *n* is a positive integer and denotes the sum of the divisors of *n*, then *n* is perfect if , abundant if and deficient if . *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 is given.

**[1]**Stan Benkoski,*Problems and Solutions: Solutions of Elementary Problems: E2308*, Amer. Math. Monthly**79**(1972), no. 7, 774. MR**1536794**, https://doi.org/10.2307/2316276**[2]**Paul Erdős,*Some extremal problems in combinatorial number theory*, Mathematical Essays Dedicated to A. J. Macintyre, Ohio Univ. Press, Athens, Ohio, 1970, pp. 123–133. MR**0276194****[3]**Pál Erdős,*Some remarks on number theory. III*, Mat. Lapok**13**(1962), 28–38 (Hungarian, with English and Russian summaries). MR**144871****[4]**P. Erdös, "On primitive abundant numbers,"*J. London Math. Soc.*, v. 10, 1935, pp. 49-58.**[5]**Yoichi Motohashi,*A note on the least prime in an arithmetic progression with a prime difference*, Acta Arith.**17**(1970), 283–285. MR**268131**, https://doi.org/10.4064/aa-17-3-283-285**[6]**W. Sierpiński,*Sur les nombres pseudoparfaits*, Mat. Vesnik**2(17)**(1965), 212–213 (French). MR**199147****[7]**Andreas Zachariou and Eleni Zachariou,*Perfect, semiperfect and Ore numbers*, Bull. Soc. Math. Grèce (N.S.)**13**(1972), no. 1-2, 12–22. MR**360455**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1974-0347726-9

Keywords:
Weird numbers,
pseudoperfect numbers,
primitive abundant numbers

Article copyright:
© Copyright 1974
American Mathematical Society