"David Wilson" <davidwwilson@attbi.com> wrote:

There are three differences between my computations and yours. (1) I use Perl (2) I index from zero and (3) I wrote the algorithm myself.

If you index from zero, i.e. u(0)=1, u(1)=2, then that is the same thing I did. My program is written in Common Lisp. I wrote it with enough complexity that I could easily have an error.

Any of these may account for discrepancies between our results, and since you and Brockhaus agree, I will let you be right, it isn't worth it to me to find my error.

Well, it's worth it to me if you can find it, because I want to know if I made an error. I wonder if you might have the correct sequence but some problem with the histogram, say with the boundaries of the bins or something. For that reason, I would really like to know what you calculate as the contents of bin [.5,.55). But this wish is not critical--don't bother if it's really a big problem. For my part, I'm going to rewrite my code (a) simply and slowly, just to test up to u(1999), and (b) with a slightly different method of optimization, to see if I can carry it out further.

All I was interested in was whether the observed bunching phenomenon was real, or whether I was totally off the wall.

It appears that both you and Brockhaus have confirmed the bunching with period about 21.6.

Well, I wouldn't call what I confirmed "bunching", but a periodic predominance. Maybe that's just a difference of vocabulary, and maybe there's something else here we haven't really tested for.

Do you think this is a constant?

In carrying it out to u(70000) it seemed to be a constant, but the value is more like 21.60157 as I said. Definitely between 21.6015 and 21.6016.

If so, what would cause this regular bunching?

Here's a phenomenon that might explain it. It is similar to a phenomenon seen in "Take or Break" games such as Grundy's Game. Suppose that the real numbers mod K are divided into a sparse space S and a common coset C such that for all s1,s2 in S, c1,c2 in C, we have s1+s2 and c1+c2 in S and s1+c2 in C. Suppose further that the sequence values are in C with finitely many exceptions X, a subset of S. Then we might see a situation where - almost all values in S are be prohibited because they appear as c1+c2 for more than one pair c1,c2 in the sequence, and - for any sequence value c1 in C and x1 in X, the value c1+x1 is in C and will be represented at least once as a sum of sequence values. So as long as there is no c2+x2=c1+x1, the value c1+x1 will be in the sequence. There seems to be something more going on here, because the density within C is not uniform. But possibly that's just because the buckets are too big to show the exact structure of C. Dan Hoey Hoey@AIC.NRL.Navy.Mil