We present a family of integer sequences characterizing the behavior of the quotients σ/s for a given odd natural number H, where N = H · 2^σ + 1 is a composite number and h · 2^s + 1 (h ≥ 1 odd, s, σ ∈ N) is a non-trivial divisor of N. As an application we prove a generalization of the primality theorem of Proth.