JHC's note suggested a couple of questions: What's the smallest ODD group-abundant number? (Ed, your table of G(N) only went up to 799.) The smallest odd abundant number is 945; we might guess the smallest odd-g-a is 2187 or 6561. I wouldn't be quite so sure that there are no G-perfect numbers. I'm curious when G(N)=1. This would be the cyclic group ZN, or (mod N). Since A|B -> G(A)<=G(B), and G(P^2)=2 (ZPxZP and (mod P^2)), N must be square-free. Moreover, G(PQ) = 2 when P|Q-1, and 1 otherwise, so we get an extra condition on the primes dividing N: PQ|N -> not(P|Q-1). Is this set of conditions sufficient? The smallest three-factor N meeting these conditions is 255 = 3.5.17, and, indeed, the table shows G(255)=1. The density of such N is <=epsilon. If I've got the predicate right, then the density eventually exceeds any ((loglogN)^K)/logN. Sparse, but not that sparse. More generally, when is G(MN)=G(M)*G(N)? If gcd(M,N)=1, we expect at least >= just from the cross products, after we've proved they are all distinct. Groupies? Rich rcs@cs.arizona.edu