Abstract

If the natural number n has the canonical form p1a1p2a2…prar then d=p1b1p2b2…prbr is said to be an exponential divisor of n if bi|ai for i=1,2,…,r. The sum of the exponential divisors of n is denoted by σ(e)(n). n is said to be an e-perfect number if σ(e)(n)=2n; (m;n) is said to be an e-amicable pair if σ(e)(m)=m+n=σ(e)(n); n0,n1,n2,… is said to be an e-aliquot sequence if ni+1=σ(e)(ni)−ni. Among the results established in this paper are: the density of the e-perfect numbers is .0087; each of the first 10,000,000e-aliquot sequences is bounded.