TITLE: Find smallest number $n\gt 1$ for which the sum of the $q$th power of its digits is $n$
QUESTION [16 upvotes]: 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 ?

REPLY [4 votes]: OEIS does know this sequence; it's http://oeis.org/A014576.
EDIT: As DJC points out in a comment, I was somewhat mistaken. To make amends, I note that these numbers are known under many names: narcissistic numbers, Armstrong numbers, and, my favorite, pluperfect digital invariant numbers. Searching under these terms will turn up many documents. It appears to have been proved that there are exactly 88 such numbers, and that the largest permissible value of $q$ is 39.