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We define a function f(n)=(n1)^(n1)+(n2)^(n2)+..., (nk)^(nk), where{ n1, n2, ..., nk } is the list of the digits of a natural number n. If we start with a natural number n, and repeatedly apply the function f, then we can generate a sequence {n, f(n), f(f(n)), f(f(f(n))), …}.
For any natural number n the sequence {n, f(n), f(f(n)), f(f(f(n))), …} eventually enters into one of 8 loops. The fact that there are only 8 loops including 2 fixed points is very curious. This fact was found by 7 high school students, and was proved using mathematics and calculation by Mathematica.
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