2 To The 2 To The 2 To The 2 To The Ad Infinitum

2,22,222,2222,\large 2,2^2,2^{2^2},2^{2^{2^2}},\ldots The sequence above can be recursively defined by a1=2,ak+1=2aka_1=2, a_{k+1}=2^{a_{k}}

Given a positive integer nn, a new sequence is created by dividing each term {ak}\{a_k\} by nn and writing down the remainder of the division. We denote this new sequence by {bk}\{b_{k}\}.

For a given nn, it is known that the terms of {bk}\{b_{k}\} eventually become constant. Let f(n)f(n) denote the index at which the constant values begin, i.e. f(n)f(n) is the smallest number ii for which bi1bi=bi+1=bi+2= b_{i-1} \neq b_i = b_{i+1} = b_{i+2} = \cdots .

Find f(2016)+f(2015)f(2016)+f(2015).

Proving that the terms of {bn}\{b_{n}\} become constant is an old USAMO problem, which inspired this problem.

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