# Continued Fractions of Harmonic Numbers

Leroy Quet qq-quet at mindspring.com
Fri May 27 21:12:11 CEST 2005

```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

```