Jeff Shallit got interested in the question "How many distinct ways are there to factor a number?" In particular, for 12, there are a great many: 12 = 12 * 1 = 6 * 2 = 4 * 3 = 2 * 2 * 3. One could say that the "logarithm" of this problem is the question of the number of additive partitions of n, which is well-studied. He observed by extensive calculation that the number of multiplicative partitions went up with n (hardly surprising) but not very fast, and that n / M(n) seemed to achieve a maximum at n = 144. I helped him with one of the proofs that he was stuck on, and in the mathematics tradition, when he submitted the paper he put my name first because I came before him in the alphabet. Later someone wrote a paper called "A proof of the Hughes-Shallit" conjecture, so I guess we were immortalized.
I don't have a copy of the paper anymore; it appeared in the American Mathematical Montly sometime in the early 1980s.
