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Say, $N=3024$, then the unique prime representation is $2^4×3^3×7$, and we can write a new number $T(3024)=22223337$. Repeating, we have $3×61×121439$ and $T^2(3024)=361121439$, again $3×7×43×399913$, giving $T^3(3024)=3743399913$. etc.  \begin{equation}  \begin{array}{|c|c|c|c|c|} \begin{array}{|c|c|c|c|c|C|}  \hline  N & T(N) & T^2(N) & T^3(N) & T^4(N) \\ & T^5(N) & T^6(N) & T^7(N) & T^8(N)\\  \hline  1 & 1 \\  2 & 2 \\  3 & 3 \\  4 & 22 2|2  & 211 2|11 \\  5 & 5 \\  6 & 23 2|3  \\ 7 & 7 \\  8 & 222 2|2|2  & 2337 2|3|3|7  & 31941 3|19|41  & 33371313 \\ 3|3|3|7|13|13 & 3|11123771 & 7|149|317|941 & 229|31219729 & 11|2084656339\\  \hline  \end{array}  \end{equation}