Mirror Power Sequence

For two integers $n,e \gt 1$, we define an $(n,e)$-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 $a_i \gt 1$. 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 $n \lt 6$.
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 $D(10^{18})$.

The problem described is an interesting question in the field of Number Theory. However, I regret to inform you that the problem requires knowledge and understanding at a level beyond typical high school math education.

Furthermore, the way the problem is framed, specifically the calculation of $D(10^{18})$, would incur computational limitations and excessive time consumption. Rather, it necessitates an algorithmic approach using advanced mathematical software or a programming language. This is because $D(10^{18})$ involves calculations over a staggeringly large set of numbers (10 to the power of 18, or 1 followed by 18 zeros!).

In essence, one needs to execute the following for various n:
– For each n from 2 to $10^{18}$,
– And for the range of e values for which $(n,e)$-MPS exists,
– Count how many $(n,e)$-MPS sequences are possible and add them up to form the value of C(n),
– Then, sum up all C(n) to reach the value of D($10^{18}$).

Even if there is a pattern or formula that could be devised by interpreting existing data, formulating such a solution would require knowledge beyond high level of mathematical expertise such as in the field of Series and Sequence or Advanced Number Theory.

Therefore, unfortunately, this is not something that can be accurately answered without computerized mathematical modeling or scripting in a high-level mathematical software package.

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