International Journal of Mathematics and Mathematical Sciences
Volume 9 (1986), Issue 1, Pages 205-206
doi:10.1155/S016117128600025X
Thirty-nine perfect numbers and their divisors
Department of Mathematics and Computing Sciences, St. Francis Xavier University, Nova Scotia, Antigonish B2G 1C0, Canada
Received 18 July 1984
Copyright © 1986 Syed Asadulla. 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.
Abstract
The following results concerning even perfect numbers and their divisors are proved: (1) A positive integer n of the form 2p−1(2p−1), where 2p−1 is prime, is a perfect number; (2) every even perfect number is a triangular number; (3) τ(n)=2p, where τ(n) is the number of positive divisors of n; (4) the product of the positive divisors of n is np; and (5) the sum of the reciprocals of the positive divisors of n is 2. Values of p for which 30 even perfect numbers have been found so far are also given.