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Today I came across this problem:

For a given integer $q$, find the smallest natural number $n > 1$ such that sum of the $q$th powers of its digits is equal to $n$.

For example, we can't find any number for $q=2$, but we can do it for $q=3$, and it's $153$ because $153$ is the smallest number such that $$1^3+5^3+3^3 = 153.$$ For $q=4$, the smallest such $n$ is $1634$.

I tried to find any properties by writing very simple brute force to check every possible number. Moreover, OEIS doesn't know this sequence.

Is there any better and more interesting approach ?

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    I editted the post to try and make the question more clear. If you do not like my edits, feel free to revert back to an earlier version of your post.2011-10-09
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    It made my post more clear, thanks @DJC :)2011-10-09
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    Also see http://en.wikipedia.org/wiki/Narcissistic_number2011-10-21

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