More About Munchausen Numbers

On September 22nd 2022, I made a post titled "What's Special About 3435?" in which it was revealed that 3435 was the only Munchausen number in base 10, apart from the trivial case of 1. It has the property that $3^3+4^4+3^3+5^5=3435$. The convention $0^0=1$ is being applied whenever a zero is encountered in a number. Using this convention, Figure1 shows the Munchausen numbers in bases from 2 to 10.

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Let's look at the Munchausen numbers for base 4:$$\begin{gathered} {29}_{10}={131}_4=1^1+3^3+1^1 \\ {55}_{10}={313}_4=3^3+1^1+3^3 \end{gathered}$$Getting back to Munchausen numbers in base 10, I got to thinking about numbers that differed by only 1 under the ${\text{digit}}^{\text{digit}}$ sum. This yielded an interesting result in the range up to 100 million (permalink):

32 --> 31

3153 --> 3152

6255 --> 6254

870206 --> 870205

1647371 --> 1647370

1647372 --> 1647373

We can see that 1647371 gives a result that is one below the number while the next consecutive number 1647372 gives a result that is one above the number. The result is that the averages are the same:$$\frac{1647371+1647372}{2}=\frac{1647370+1647373}{2}=1647371.5$$I thought that was a pretty interesting result. 

Another variation is to consider squares and cubes of numbers and beyond. By this I mean what numbers have the property that:$${\text{number}}^n=\sum ({\text{digit}}^{\text{digit}})$$where $n$ can equal 1, 2, 3 etc.

So far we've only considered the case of $n=1$. What about if $n=2$? In the range up to ten million, we find only the numbers 1, 2 and 216 satisfying the condition (permalink):

$$\begin{array}{rl}{\mathbf{\text{1}}}^2& =1^1\\ {\mathbf{\text{2}}}^2& =2^2\\ {\mathbf{\text{216}}}^2& =2^2+1^1+6^6\end{array}$$What about cubes? Here we find, again in the range up to ten million, that only 1, 3, 36 and 729 qualify (permalink):$$\begin{array}{rl}{\mathbf{\text{1}}}^3& =1^3\\ {\mathbf{\text{3}}}^3& =3^3\\ {\mathbf{\text{36}}}^3& =3^3+6^6\\ {\mathbf{\text{729}}}^3& =7^7+2^2+9^9\end{array}$$Perhaps we could term such numbers Munchausen numbers of the second order (for squares), Munchausen numbers of the third order (for cubes) and so on.  I thought this was an interesting variation on the original theme. Obviously one could go on and look at higher orders but I'll stop at 3.