$2023$ is the smallest nontrivial positive integer (that is, not $1$) in base $10$ that has the property that the sum of its digits times the square of the sum of the squares of its digits is itself (that is, $(2 + 0 + 2 + 3) \cdot (2^2 + 0^2 + 2^2 + 3^2)^2 = 2023$. What is a general way to find the smallest such nontrivial positive integer for any base?
Base $2.$ No such nontrivial positive integer exists.