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

Pomerance, Carl; Yang, Hee-Sung

doi:10.1090/S0025-5718-2013-02775-5

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.

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