I have grave doubts about the possibility of this problem having a solution as stated. I have (vaguely) heard of Ulam, but have never studied any of his books so I can't say for sure, but certainly within the limits of my knowledge of Physics and my creativity I can't think of a way to construct such a series of non-zero masses.
Until someone can post a workable solution I'm going to have to say that I think it is impossible.
Perhaps you could try posting your question here. If they do manage to come up with a solution I'd be interested in hearing about it.