Energetic Numbers Revisited

In June of 2024, I posted about Energetic Numbers which are numbers that can be split into two or more parts and these parts, when raised to appropriate integer powers and added together, equal the original number. An example is 27476:$$ 27476 =27^3 + 4^6 + 7^4 + 6^4 $$In that post I made reference to a text file that listed the first 10000 energetic numbers but did not show how the numbers could be split and recombined.

Fortunately, I was able to get Gemini to create a SageMath program (permalink) that accomplishes this task whenever it is possible. I was prompted to attempt this because 27892, the number associated with my diurnal age today, is one of the numbers listed in the previously mentioned text file. For this number we have:$$27892=27^3 + 8 + 9 + 2^{13}$$Here is a list of some of the upcoming energetic numbers. Notice the run of consecutive numbers from 28260 to 28269 (permalink):

\(27892 = 27^3 + 8 + 9 + 2^{13}\) 
\(27923 = 2^{13} + 7^5 + 9^3 + 2^3 + 3^7\)
\(27962 = 27^3 + 9^2 + 6 + 2^{13}\)
\(27972 = 27^3 + 97 + 2^{13}\)
\(27984 = 2^9 + 7^5 + 9^4 + 8 + 4^6\)
\(28132 = 2^{12} + 8^4 + 1 + 3^9 + 2^8\)
\(28160 = 2^{11} + 8^3 + 160^2\)
\(28203 = 2^9 + 8 + 20^3 + 3^9\)
\(28224 = 28^3 + 2^{11} + 2^7 + 4^6\)
\(28228 = 2^{14} + 82^2 + 2^10 + 8^4\)
\(28243 = 28^3 + 2^3 + 4^6 + 3^7\)
\(28245 = 2^{13} + 8^3 + 2^5 + 4^7 + 5^5\)
\(28260 = 2^{14} + 8^4 + 2^2 + 6^5 + 0\)
\(28261 = 2^{14 }+ 8^4 + 2^2 + 6^5 + 1\)
\(28262 = 2^{14 }+ 8^4 + 2^2 + 6^5 + 2\)
\(28263 = 2^{14} + 8^4 + 2^2 + 6^5 + 3\)
\(28264 = 2^{14 }+ 8^4 + 2^2 + 6^5 + 4\)
\(28265 = 2^{14 }+ 8^4 + 2^2 + 6^5 + 5\)
\(28266 = 2^{14} + 8^4 + 2^2 + 6^5 + 6\)
\(28267 = 2^{14} + 8^4 + 2^2 + 6^5 + 7\)
\(28268 = 2^{14} + 8^4 + 2^2 + 6^5 + 8\)
\(28269 = 2^{14} + 8^4 + 2^2 + 6^5 + 9\)

27892 also has the property that it can be split into two parts and the two parts, when added together, form a palindromic prime with a prime index:$$ \begin{align} 27892 &\rightarrow 27 + 892 \\  &=919          \\ &=\text{prime}(157) \end{align}$$