International Journal of Mathematics and Mathematical Sciences
Volume 11 (1988), Issue 2, Pages 343-349

Some results concerning exponential divisors

Peter Hagis Jr.

Mathematics Department, Temple University, Philadelphia 19122, PA, USA

Received 18 November 1986; Revised 9 February 1987

Copyright © 1988 Peter Hagis. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


If the natural number n has the canonical form p1a1p2a2prar then d=p1b1p2b2prbr 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.