For a positive integer r, define j_r to be the smallest positive integer n satisfying n! > n^r–1. In this paper we prove j_r+1 ∈ {j_r + 1, j_r + 2}, which leads us to explore the set of positive integers r for which j_r+1 = j_r + 2. We prove this set has the same density as the prime numbers. The sequence j_r was introduced by Carlson, Goedhart, and Harris in their work on factoradic happy numbers, and we prove some properties of j_r that lead to an improvement of one of their theorems.