On weird and pseudoperfect numbers
Authors:
S. J. Benkoski and P. Erdős
Journal:
Math. Comp. 28 (1974), 617623
MSC:
Primary 10A40
Corrigendum:
Math. Comp. 29 (1975), 673674.
Corrigendum:
Math. Comp. 29 (1975), 673.
MathSciNet review:
0347726
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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, http://dx.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
(43 #1942)
 [3]
Pál
Erdős, Some remarks on number theory. III, Mat. Lapok
13 (1962), 28–38 (Hungarian, with Russian and
English summaries). MR 0144871
(26 #2412)
 [4]
P. Erdös, "On primitive abundant numbers," J. London Math. Soc., v. 10, 1935, pp. 4958.
 [5]
Yoichi
Motohashi, A note on the least prime in an arithmetic progression
with a prime difference, Acta Arith. 17 (1970),
283–285. MR 0268131
(42 #3030)
 [6]
W.
Sierpiński, Sur les nombres pseudoparfaits, Mat. Vesnik
2 (17) (1965), 212–213 (French). MR 0199147
(33 #7296)
 [7]
Andreas
Zachariou and Eleni
Zachariou, Perfect, semiperfect and Ore numbers, Bull. Soc.
Math. Grèce (N.S.) 13 (1972), no. 12,
12–22. MR
0360455 (50 #12905)
 [1]
 S. J. Benkoski, "Elementary problem and solution E2308," Amer. Math. Monthly, v. 79, 1972, p. 774. MR 1536794
 [2]
 P. Erdös, "Some extremal problems in combinatorial number theory," Mathematical Essays Dedicated to A. J. Macintyre, Ohio Univ. Press, Athens, Ohio, 1970, pp. 123133. MR 43 #1942. MR 0276194 (43:1942)
 [3]
 P. Erdös, "Remarks on number theory. III," Mat. Lapok, v. 13, 1962, pp. 2838. (Hungarian) MR 26 #2412. MR 0144871 (26:2412)
 [4]
 P. Erdös, "On primitive abundant numbers," J. London Math. Soc., v. 10, 1935, pp. 4958.
 [5]
 Y. Motohashi, "A note on the least prime in an arithmetic progression with a prime difference," Acta Arith., v. 17, 1970, pp. 283285. MR 42 #3030. MR 0268131 (42:3030)
 [6]
 W. Sierpinski, "Sur les nombres pseudoparfaits," Mat. Vesnik, v. 2(17), 1965, pp. 212213. MR 33 #7296. MR 0199147 (33:7296)
 [7]
 A. & E. Zachariou, "Perfect, semiperfect and Ore numbers," Bull. Soc. Math. Grèce, v. 13. 1972, pp. 1222. MR 0360455 (50:12905)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197403477269
PII:
S 00255718(1974)03477269
Keywords:
Weird numbers,
pseudoperfect numbers,
primitive abundant numbers
Article copyright:
© Copyright 1974 American Mathematical Society
