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

We study the distribution of solutions n to the congruence sigma(n) equivalent to a (mod n). After excluding obvious families of solutions, we show that the number of these n <= x is at most x(1/2+o(1)), as x -> infinity, uniformly for integers a with vertical bar a vertical bar = x(1/4). As a concrete example, the number of composite solutions n <= x to the congruence sigma(n) equivalent to 1 (mod n) is at most x(1/2+o(1)). These results are analogues of theorems established for the Euler phi-function by the third-named author.