A generalization of Everett's result on the Collatz 3x + 1 problem

It is a well-known conjecture that given m ε N, the set of natural numbers, the sequence {m_n}^∞_n−0, defined by the iterative formula m₀ = m,

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has some iterate m_j = 1. It is shown in this paper that for any k ε N, “almost every” natural number m greater than unity has k iterates less than m.