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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Variant of a theorem of Erdős on the sum-of-proper-divisors function
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by Carl Pomerance and Hee-Sung Yang PDF
Math. Comp. 83 (2014), 1903-1913 Request permission

Abstract:

In 1973, Erdős proved that a positive proportion of numbers are not of the form $\sigma (n)-n$, the sum of the proper divisors of $n$. We prove the analogous result where $\sigma$ is replaced with the sum-of-unitary-divisors function $\sigma ^*$ (which sums divisors $d$ of $n$ such that $(d, n/d) = 1$), thus solving a problem of te Riele from 1976. We also describe a fast algorithm for enumerating numbers not in the form $\sigma (n)-n$, $\sigma ^*(n)-n$, and $n-\varphi (n)$, where $\varphi$ is Euler’s function.
References
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Additional Information
  • Carl Pomerance
  • Affiliation: Department of Mathematics, 6188 Kemeny Hall, Dartmouth College, Hanover, New Hampshire 03755
  • MR Author ID: 140915
  • Email: carl.pomerance@dartmouth.edu
  • Hee-Sung Yang
  • Affiliation: Department of Mathematics, 6188 Kemeny Hall, Dartmouth College, Hanover, New Hampshire 03755
  • Email: hee-sung.yang.12@dartmouth.edu
  • Received by editor(s): June 14, 2012
  • Received by editor(s) in revised form: July 31, 2012, and December 10, 2012
  • Published electronically: October 29, 2013
  • © Copyright 2013 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Math. Comp. 83 (2014), 1903-1913
  • MSC (2010): Primary 11A25, 11Y70, 11Y16
  • DOI: https://doi.org/10.1090/S0025-5718-2013-02775-5
  • MathSciNet review: 3194134