Continued Fractions of Harmonic Numbers

[Leroy Quet]

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,...

Is this sequence in the EIS?

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?

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.

Sequence starts: 1,...

I do not know if this sequence has every term defined either.

thanks,
Leroy Quet