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Mirror Power Sequence

Problem 617 Published on Saturday, 30th December 2017, 04:00 pm; Solved by 224

For two integers $n,e > 1$, we define a $(n,e)$-MPS (Mirror Power Sequence) to be an infinite sequence of integers $(a_i)_{i\ge 0}$ such that for all $i\ge 0$, $a_{i+1} = min(a_i^e,n-a_i^e)$ and $a_i > 1$.
Examples of such sequences are the two $(18,2)$-MPS sequences made of alternating $2$ and $4$.

Note that even though such a sequence is uniquely determined by $n,e$ and $a_0$, for most values such a sequence does not exist. For example, no $(n,e)$-MPS exists for $n < 6$.

Define $C(n)$ to be the number of $(n,e)$-MPS for some $e$, and $\displaystyle D(N) = \sum_{n=2}^N {C(n)}$.
You are given that $D(10) = 2$, $D(100) = 21$, $D(1000) = 69$, $D(10^6) = 1303$ and $D(10^{12}) = 1014800$.

Find $D(10^{18})$.