First Occurrence Of Floor(m/d(m))

[David Wilson]

In these cases, empirical evidence sometimes gives us a feel for what the 
answer might be.

For example, with Goldbach's conjecture, if we look at the graph of A045917, 
we see the general upward trend of the number of Goldbach partitions of 2n. 
Unfortunately, our good friend Tony Noe has not yet added the b-file, or 
else I'm sure we would the trend would be much more convincing. In this 
case, the visual evidence is supported by statistical estimates of the 
number of Goldbach partitions of 2n there should be. This gives us a warm 
fuzzy that Goldbach's conjecture should be true.

Obviously, such visual evidence is not proof can be misleading (vis Merten's 
conjecture), but it can give us a working opinion.

I suggest you graph the number f(n) = |{m: floor(m/d(m)) = n}|. It may give 
you insight into what to think.

Also, for a few small m, you might want to look at the m for which 
floor(m/d(m)) = n on the off chance that there might be an exploitable 
pattern (though I'm dubious of a simple one).

> The first question is harder to answer. I still believe that there
> exists n for which floor(m/d(m))=n has no solutions but a smallest
> such n may be quite large.
>
> Max