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I noticed (with the help of the Instagram algorithm) that there are certain numbers, that when raised to a certain power, have a result whose digits add up to the original base.

Some examples of this are $9^2, 7^4, 36^5,$ so on and so forth.

I was wondering if there was a list somewhere of every number that has this property, so I could try to find some sort of pattern to this. If not, I bet I could probably make something in google sheets to generate a list of numbers that do this.

I also made a working spreadsheet that gave lots of results, I'll post my findings once I collect them and make them look a bit nicer.

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  • $\begingroup$ For some basic information about writing mathematics at this site see, e.g., here, here, here and here. $\endgroup$
    – Another User
    Commented Aug 3, 2022 at 16:00
  • $\begingroup$ appreciate it thanks $\endgroup$
    – adam3560
    Commented Aug 3, 2022 at 16:04
  • $\begingroup$ oeis.org/A023052 (OEIS is always a good place to start looking for sequences). However, that doesn't include 36, so I may have to dig deeper or I found an error in OEIS $\endgroup$
    – Barry Carter
    Commented Aug 3, 2022 at 16:07
  • $\begingroup$ @BarryCarter That sequence is different than the one here. A023052 are numbers that are a sum of the digits which are themselves raised to powers like $153=1^3+5^3+3^3$, not a sum of the digits of a power of the number like $9^2=81$ with $9=8+1$. $\endgroup$
    – Merosity
    Commented Aug 3, 2022 at 18:07
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    $\begingroup$ Messing around on Python, I found some larger examples, such as $201384^{8479}$ has a digit sum of $201384$. $\endgroup$
    – paw88789
    Commented Aug 4, 2022 at 17:39

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I wrote a couple of blog posts about these a while back and included a list of the first few dozen examples, up to $265^{28}$.

Some numbers with this property are sometimes called Dudeney numbers.

OEIS has more information, including a list of the first 1138 examples.

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