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Let K be an arbitrary positive number. We define a function
f(n) = Mod[(n1)^(n1) + (n2)^(n2) + ..., (nk)^(nk),K],
where { n1,n2, ..., nk } is the list of the digits of a natural number n. If we repeatedly apply the function f, then we can generate a sequence {n, f(n), f(f(n)), f(f(f(n))), .....}.
If we choose a proper number K, then for any natural number n the sequence {n, f(n), f(f(n)), f(f(f(n))), ...} eventually converges to 1.
This curious fact is thoroughly explored in Mathematica.
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