Call a factorization distinct if there are no repeated factors, and call a factorization square-free if it contains no squares. For all positive integers > 1 I claim the number of each match. For example of the sixteen factorizations of 72: A 72 B 8.9 C 6.12 D 4.18 E 3.24 F 3.4.6 G 3.3.8 H 2.36 I 2.6.6 J 2.4.9 K 2.3.12 L 2.3.3.4 M 2.2.18 N 2.2.3.6 O 2.2.2.9 P 2.2.2.3.3 seven (GILMNOP) have repeated factors, and seven (BDFHJLO) contain 4, 9 or 36. Do you know any citation for this? (Assuming it's in fact true!<;-) Also, I'd like to see your proofs, to see how they differ from mine. Particularly interesting would be an explicit construction mapping the factorizations 1-1. (At first I thought this was easy--just split q^2 into q.q--but it's unclear to me what the images of J or P, say, ought to be). Thanks!