  Let $f(n)$ be the sum of the prime divisors of $n$, counted with
multiplicity; thus $f(2020)$ $= f(2^2 \cdot 5 \cdot 101) = 110$. Ruth-Aaron
numbers, or integers $n$ with $f(n)=f(n+1)$, 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
Erd\"os and Pomerance in 1978. In this paper, we generalize their results in
two directions: by raising prime factors to a power and allowing a small
difference between $f(n)$ and $f(n+1)$. We prove that the number of integers up
to $x$ with $f_r(n)=f_r(n+1)$ is $O\left(\frac{x(\log\log x)^3\log\log\log
x}{(\log x)^2}\right)$, where $f_r(n)$ is the Ruth-Aaron function replacing
each prime factor with its $r-$th power. We also prove that the density of $n$
remains $0$ if $|f_r(n)-f_r(n+1)|\leq k(x)$, where $k(x)$ is a function of $x$
with relatively low rate of growth. Moreover, we further the discussion of the
infinitude of Ruth-Aaron numbers and provide a few possible directions for
future study.
