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Variant of a theorem of Erdős on the sum-of-proper-divisors function


Authors: Carl Pomerance and Hee-Sung Yang
Journal: Math. Comp. 83 (2014), 1903-1913
MSC (2010): Primary 11A25, 11Y70, 11Y16
Published electronically: October 29, 2013
MathSciNet review: 3194134
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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.


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Additional Information

Carl Pomerance
Affiliation: Department of Mathematics, 6188 Kemeny Hall, Dartmouth College, Hanover, New Hampshire 03755
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

DOI: http://dx.doi.org/10.1090/S0025-5718-2013-02775-5
Keywords: Unitary untouchable numbers, untouchable numbers, noncototients
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
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.