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 ?