Friends of 12

Abstract

A friend of 12 is a positive integer different from 12 with the same abundancy index. By enlarging the supply of methods of Ward [1], it is shown that (i) if n is an odd friend of 12, then n=m^2, where m has at least 5 distinct prime factors, including 3, and (ii) if n is an even friend of 12 other than 234, then n=2*(q^e)*(m^2), in which q is a prime greater than or equal to 29, e is a positive integer, and both q and e are congruent to 1 mod 4, and m has at least 3 distinct odd prime factors, one of which is 3, and the other, none equal to q, are greater than or equal to 29.