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If we let S(n) be the sum of the prime factors (with multiplicity) of an integer n, then we define a Ruth-Aaron Number to be any n such that S(n)=S(n+1). The main results regarding Ruth-Aaron numbers that Erdos and Pomerance proved show that the Ruth-Aaron numbers have density zero and the sum of their reciprocals converges. We extend their results by replacing the function S (which sums the prime powers of a number n) with other functions f on the prime factors. In the process we take a look at the history of analytic number theory, and review some classical results from the field.