For two integers $n,e>1$, 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 $a_i>1$.
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 $D({10}^{18})$.