On Exponentially Perfect Numbers Relatively Prime to 15

If the natural number n has the canonical form p₁^a₁p₂^a₂⋯p_r^a_r, then we say that an exponential divisor of n has the form d = p₁^b₁p₂^b₂⋯p_r^b_r, where b_i|a_i for i = 1, 2, … r. We denote the sum of the exponential divisors of n by σ^(e)(n). A natural number n is said to be exponentially perfect (or e-perfect) if σ^(e)(n) = 2n.

The purpose of this thesis is to investigate the existence of e-perfect numbers relatively prime to 15. In particular, if such numbers exist, are they bounded below? How many distinct prime divisors must they have? Several lemmas are utilized throughout the paper on route to answering these questions. Also, computer programs written in Maple are used for numerical estimates.