The Devil is in the Details

Calculus Level 4

Consider the recurrence relation $$\large f_{n+1} (x) = x^{ f_n (x) }$$, for non-negative integers $$n$$ with $$f_0 (x) = 1$$.

$\large\displaystyle \lim_{x \to 1} \frac { f_{666} (x) - 1}{x^{666} - 1}$

Let the limit above equals to $$\frac AB$$ for coprime positive integers $$A$$ and $$B$$.

Denote $$D$$ as the absolute difference between the smallest and largest prime factors of $$A+B$$.

What is the value of $$1 + 2 + 3 + \ldots + (D^2 - 1) + D^2$$?

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