Sums of Divisors and Egyptian Fractions Original Research Article

We discuss the problem of representing a natural number n as a sum of certain of its distinct positive proper (≠n) divisors. If this is possible n is called semiperfect. We present a method which leads in certain cases to a verification that all abundant numbers with prime divisors limited to a fixed set of primes are semiperfect. We then discuss the Egyptian fraction representation of positive rational numbers (i.e., the representation as a sum of reciprocals of distinct integers > 1). In particular we obtain results relating the primes occurring in such representations to the primes occurring in the denominator of the rational being represented. This leads to further results concerning the existence of semiperfect numbers with certain prime factors.