Generalizing Ruth-Aaron Numbers

A Ruth-Aaron number is such that the sum of its prime divisors, counted with multiplicity, equals the sum of the prime divisors of its successor. They have been an interest of many number theorists since the famous 1974 baseball game gave them the elegant name after two baseball stars. Many of their properties were first discussed by Erdős and Pomerance in 1978. We generalize their results in two directions: by raising prime factors to a power and allowing a small difference between the two sums. We prove the density of power Ruth-Aaron numbers is zero (we provide a precise bound in terms of iterations and powers of logarithms), and prove similar results for almost Ruth-Aaron numbers (these are pairs of integers such that the sum of their prime powers differ by a fixed, slowly growing function). Moreover, we further the discussion of the infinitude of Ruth-Aaron numbers and provide a few possible directions for future study.