Remarks on Fibers of the Sum-of-Divisors Function

Let σ $$\mbox{$\sigma$}$$ denote the usual sum-of-divisors function. We show that every positive real number can be approximated arbitrarily closely by a fraction m∕n with σ ( m ) = σ ( n ) $$\sigma (m) =\sigma (n)$$ . This answers in the affirmative a question of Erdős. We also show that for almost all of the elements v of σ ( N ) $$\sigma (\mathbf{N})$$ , the members of the fiber σ − 1 ( v ) $$\sigma ^{-1}(v)$$ all share the same largest prime factor. We describe an application of the second result to the theory of L.E. Dickson’s amicable tuples, which are a generalization of the ancient notion of an amicable pair.