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We define a multiplicative arithmetic function D by assigning D(p a) = ap a−1, when p is a prime and a is a positive integer, and, for n ≥ 1, we set D0 (n) = n and Dk (n) = D(Dk−1 (n)) when k ≥ 1. We term {Dk(n)} ∞ k=0 the derived sequence of n. We show that all derived sequences of n < 1.5 · 10 10 are bounded, and that the density of those n ∈ N with bounded derived sequences exceeds 0.996, but we conjecture nonetheless the existence of unbounded sequences. Known bounded derived sequences end (effectively) in cycles of lengths only 1 to 6, and 8, yet the existence of cycles of arbitrary length is conjectured. We prove the existence of derived sequences of arbitrarily many terms without a cycle.

Year: 2011

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