Abstract

For any k ≥ 0, all prime n satisfy the congruence nσ_k (n) ≡ 2 mod φ(n). We show that this congruence in fact characterizes the primes, in the sense that it is satisfied by only finitely many composite n. This characterization generalizes the results of Lescot and Subbarao for the cases k = 0 and k = 1. For 0 ≤ k ≤ 14, we enumerate the composite n satisfying the congruence. We also prove that any composite n which satisfies the congruence for some k satisfies it for infinitely many k.