On weird and pseudoperfect numbers
HTML articles powered by AMS MathViewer
- 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.References
- Stan Benkoski, Problems and Solutions: Solutions of Elementary Problems: E2308, Amer. Math. Monthly 79 (1972), no. 7, 774. MR 1536794, DOI 10.2307/2316276
- 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
- Pál Erdős, Some remarks on number theory. III, Mat. Lapok 13 (1962), 28–38 (Hungarian, with English and Russian summaries). MR 144871 P. Erdös, "On primitive abundant numbers," J. London Math. Soc., v. 10, 1935, pp. 49-58.
- Yoichi Motohashi, A note on the least prime in an arithmetic progression with a prime difference, Acta Arith. 17 (1970), 283–285. MR 268131, DOI 10.4064/aa-17-3-283-285
- W. Sierpiński, Sur les nombres pseudoparfaits, Mat. Vesnik 2(17) (1965), 212–213 (French). MR 199147
- 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
Additional Information
- © Copyright 1974 American Mathematical Society
- Journal: Math. Comp. 28 (1974), 617-623
- MSC: Primary 10A40
- DOI: https://doi.org/10.1090/S0025-5718-1974-0347726-9
- MathSciNet review: 0347726