Cont Frac of Harmonic Numbers

[Leroy Quet]

(Posted to seqfan and sent to Benoit Cloitre, and perhaps later to be 
posted to sci.math) 

Just browsing the OEIS, I came across this old sequence of mine (which 
has had some comments added since I originally sent the sequence to the 
EIS).

http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?A
num=A055573

*the sequence is the number of terms in the continued fraction of
sum{k=1 to n} 1/k 
(the k_th harmonic number).

What has intrigued me is the comment from Benoit Cloitre:

"lim n -> infinity a(n)/n = C = 0.84... "

How is it proved that C exists? What are more digits of C? And, of 
course, does C have a closed-form?


And an old question related to the sequence:

Are there only a finite number of times each particular integer appears 
in the sequence?
(It is very well known that the integer 1 appears only appears once, as 
an example.)


And finally,
what are the asymptotics of the related sequence:

http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?A
num=A058027

(the SUM of the terms in the continued fractions of the harmonic numbers)


thanks,
Leroy Quet