Continued Fractions of Harmonic Numbers

[Joshua Zucker]

On 5/27/05, Leroy Quet <qq-quet at mindspring.com> wrote:
> I am wondering about the sequence where the nth term is the lowest
> positive m where the simple continued fraction of
> sum{k=1 to m} 1/k
> has exactly n terms.
> 
> I get that the sequence starts 1, 2, 3, 6, 5,...

I get 1,2,3,6,5,7,8,10,14,9,18 but a(12) may not exist.
Is my algorithm working OK?  My spot checks suggest that the program is good.
 
> Is this sequence in the EIS?

I don't find it.

> Is there a term defined for every n? ie - is every number of terms
> represented among the continued fractions of harmonic numbers?
> Or are there some n's where no CF of any harmonic number has exactly n
> terms?

I find many numbers that don't seem to be present, and judging by the
rate of increase I doubt it'll come back and pick things up later. 
There's no 12, 13, 14, 22, 27, 28, ...
among the first thousand harmonic series partial sums, anyway.

> 
> Another related sequence I wonder about:
> 
> If there are a finite number of m's where, for every n, the mth harmonic
> number has exactly n terms in its CF (if there are no n's where an
> infinite number of harmonic numbers has exactly n terms in their CFs),
> then we can form the sequence of the *highest* integer m where the mth
> harmonic number has exactly n terms.

Again, the rate of growth suggests to me that the sequence never comes
back down to the small numbers, so I conjecture the obvious stuff,
1,4,3,6,5,7,11,... (also not in EIS)

Same conjecture, undefined for 12,13,14,22,27,28... (also doesn't seem
to be in EIS).

If someone wants me to get the program to spit out more terms of all
these sequences, let me know.

--Joshua Zucker