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

Abstract

In 1973, Erdős proved that a positive proportion of numbers are not of the form σ ( n ) − n \sigma (n)-n , the sum of the proper divisors of n n . We prove the analogous result where σ \sigma is replaced with the sum-of-unitary-divisors function σ ∗ \sigma ^* (which sums divisors d d of n n such that ( d , n / d ) = 1 (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 σ ( n ) − n \sigma (n)-n , σ ∗ ( n ) − n \sigma ^*(n)-n , and n − φ ( n ) n-\varphi (n) , where φ \varphi is Euler’s function.