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On asymptotic properties of aliquot sequences


Author: P. Erdős
Journal: Math. Comp. 30 (1976), 641-645
MSC: Primary 10A20
DOI: https://doi.org/10.1090/S0025-5718-1976-0404115-8
MathSciNet review: 0404115
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Abstract: Put $ {s^{(1)}}(n) = \sigma (n) - n,\sigma (n) = {\Sigma _{d/n}}d$. $ {s^k}(n) = {s^{(1)}}({s^{(k - 1)}}(n))$. In this note we prove that for every k the density of integers satisfying

$\displaystyle {s^k}(n) = (1 + \sigma (1))n{((\sigma (n) - n)/n)^k}$

is 1. Several unsolved problems are stated.

References [Enhancements On Off] (What's this?)

  • [1] E. CATALAN, Bull. Soc. Math. France, v. 16, 1887/88, pp. 128-129.
  • [2] L. E. DICKSON, "Theorems and tables on the sum of the divisors of a number," Quart. J. Math., v. 44, 1913, pp. 264-296.
  • [3] RICHARD K. GUY & J. L. SELFRIDGE, "Interim report on aliquot series," Proc. Manitoba Conf. Numerical Math. (Univ. Manitoba, Winnipeg, Man., 1971), Dept. Comput. Sci., Univ. Manitoba, Winnipeg, Man., 1971, pp. 557-580. MR 49 #194. MR 0335412 (49:194)
  • [4] RICHARD K. GUY & J. L. SELFRIDGE, "What drives an aliquot sequence?," Math. Comp., v. 29, 1975, pp. 101-107. MR 0384669 (52:5542)
  • [5] RICHARD K. GUY, D. H. LEHMER, J. L. SELFRIDGE & M. C. WUNDERLICH, "Second report on aliquot sequences," Proc. Third Manitoba Conf. Numerical Math. (Winnipeg, Man., 1973), Utilitas Math., Winnipeg, Man., 1974, pp. 357-368. MR 50 #4455. MR 0351967 (50:4455)
  • [6 H] J. J. te RIELE, "A note on the Catalan-Dickson conjecture," Math. Comp., v. 27, 1973, pp. 189-192. MR 48 #3869. MR 0325522 (48:3869)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1976-0404115-8
Article copyright: © Copyright 1976 American Mathematical Society

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