Cont Frac of Harmonic Numbers

[benoit]

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

That was experimental result :

For m  integer >=1 let a(n,m)= number of elements in continued fraction  
for sum(i=1,n,1/i^m)

I noticed that for fixed m,  limit n-->infty  a(n,m)/n exists = C(m)  
(cf. A070985)

I noticed also C(m) is closed from m*C(1)=m*C. Does C(m)=m*C ?

If true, constant C=0.84... would be more interesting.

Some values :

m----a(10000,m)/(10000*m)

1-----0.835599
2-----0.841299
3-----0.838599
4-----0.845199
5-----0.843979
6-----0.839083
7-----0.849357
8-----0.838137

Seems support the conjecture.

> 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
>
>


Should be asymptotic to D*n^(3/2)  D=0.4....


BC
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