On the maximum number of distinct factors of a binary string

In this note we prove that a binary string of lengthn can have no more than (2^{k + 1} - 1 + left( {mathop {n - k + 1}limits_2 } right)) distinct factors, wherek is the unique integer such that 2^k + k - 1 ≤ n < 2^k+1 + k. Furthermore, we show that for eachn, this bound is actually achieved. The proof uses properties of the de Bruijn graph.