ON CONGRUENCES OF THE FORM σ(n) ≡ a ( mod n)

We study the distribution of solutions n to the congruence σ(n) ≡ a ( mod n). After excluding obvious families of solutions, we show that the number of these n ≤ x is at most x½+o(1), as x → ∞, uniformly for integers a with ∣a∣ ≤ x¼. As a concrete example, the number of composite solutions n ≤ x to the congruence σ(n) ≡ 1 ( mod n) is at most x½+o(1). These results are analogues of theorems established for the Euler ϕ-function by the third-named author.