Fourth Loneliest Number

Algebra Level 4

Consider the recurrence relation \(f_n (x) = x^{f_{n-1}(x)} \) with \(f_0(x) = 16 \). Find the number of integer solutions of \(n\) in between 1 and 1000 inclusive such that \(f_n(\sqrt[4]2) = 16 \).

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