On the maximum number of distinct factors of a binary string

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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 2k + k - 1 ≤ n < 2k+1 + k. Furthermore, we show that for eachn, this bound is actually achieved. The proof uses properties of the de Bruijn graph.