For integers s ≥ 1 and n ≥ 2, we define the function T_s as follows: T_s(n) = T_s(p^a q^b ... r^c) = ap^s + bq^s + ... + cr^s. Thus T_s(n) is the sum of the s^th powers of the prime factors of n, counted according to multiplicity of the prime factors. The set T^*(s) is defined as { n : T_s(n) = n }, and we let a(s) be the smallest element in T^*(s). We consider several natural questions. Is the set T^*(s) empty, finite or infinite for some particular values of s? Suppose y is a prime power, say y = p^m. Is it possible that y = T_s(y) for some s? What is the smallest element a(s) in the set T^*(s)? The answer for the last question is documented, but only for certain small values of s in the title sequence, a(s), for s = 1, 2, ..., namely sequence A068916 in the Online Encyclopedia of Integer Sequences. It begins 2, 16, 1096744, 3125, ... . Some sets T^*(s) are known to have one or two elements, and T^*(1) is infinite. Some sets have prime powers. In fact, infinitely often T^*(s) contains p^y for some power y and prime p. For example T^*(1) contains 3²⁷, which may be the value of a(24). The set T^*(3) contains six known elements, and none of these are prime powers. We prove T^*(3) does not contain any prime powers at all. Curiously, every known member of T^*(s) for any value of s, except s = 3, is in fact a prime power. We also briefly discuss algorithms and functions related to T_s(n).