Sums of divisors, perfect numbers, and factoring

Let <italic>N</italic> be a positive integer, and let &sgr;(<italic>N</italic>) denote the sum of the positive integral divisors of <italic>N</italic>. We show computing &sgr;(<italic>N</italic>) is equivalent to factoring <italic>N</italic> in the following sense: there is a random polynomial time algorithm that, given &sgr;(<italic>N</italic>), produces the prime factorization of <italic>N</italic>, and &sgr;(<italic>N</italic>) can be easily computed given the factorization of <italic>N.</italic> We show that the same sort of result holds for &sgr;<subscrpt>k</subscrpt>(<italic>N</italic>), the sum of the <italic>k</italic>-th powers of divisors of <italic>N.</italic> We give three new examples of problems that are in Gill's complexity class BPP: {<italic>perfect numbers</italic>}, {<italic>multiply perfect numbers</italic>}, and <italic>{amicable pairs}.</italic> These are the first “natural” candidates for BPP - RP.