Let n be a positive integer. We define
We then form the sequence obtained by iterating T: n. T(n), T(T(n), T(T(T(n))). … For instance, starting with n = 7, we have 7, 11, 17, 26, 13, 20, 10, 5, 8, 4, 2, 1, 2, 1, 2, 1, …. A well-known conjecture, sometimes called the Collar conjecture, asserts that the sequence obtained by iterating T always reaches the integer 1 no matter which positive integer n begins the sequence.
Verify that there is a term in the sequence obtained by iterating T, starting with the positive integer n that is less than n for all positive integers n with 2≤ n ≤ 100. (Hint: Begin by considering sets of positive integers for which it is easy to show that this is true.)
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