Does there exist a number $n >1$ such that $n = s(n)^{s(n)}$?
Solution 1:
I considered the prime factorization. Since n and s(n) must consist of same selection of primes, we must focus on numbers having small number of prime divisors with large powers. For example if we pick 2 and 3, our number n must at least (2.3)^6 since sum of digits (if it can) is at least 2.3.