Continued Fractions of Harmonic Numbers

[Joshua Zucker]

Has someone (Leroy, RGWv, ?) submitted the other three relevant sequences:
  1) a(n) = Length of continued fraction for 1/1 + 1/2 + 1/3 + ... + 1/n
  2) b(n) = Smallest m such that the length of the continued fraction is n
  3) c(n) = Largest m such that the length of the continued fraction is n.

If nobody else has, I'm happy to, but I think Leroy deserves most of
the credit since all three of these sequences were his ideas to begin
with.

Thanks,
--Joshua Zucker

On 5/30/05, zak seidov <zakseidov at yahoo.com> wrote:
> Just submitted A107921
> is related
> (not so much?)
> with this Continued Fractions of Harmonic Numbers
> thread,
> Zak
> 
> %I A107921
>  %S A107921
> 1,1,2,5,4,7,5,9,11,9,14,18,24,27,18,13,26,26,22,32,31,40,28,40,40,48,48,45,45,51,58,54,54,74,55,54
>  %N A107921 Smallest m such that the length of
>  continued fraction for sum 1/n+1/1+1/2+...+1/m is n.
>  %H A107921 Eric W. Weisstein,  <a
> href="http://mathworld.wolfram.com/ContinuedFraction.html">Continued
>  Fraction</a>.
>  %e A107921 a(4) = 5 because sum 1/4 + (1/1 +...+1/5)
>  = 38/15, and continued fraction for 38/15 has terms
> {2,1,1,7} and length 4.
>  %t A107921
> bb={};Do[s=1/n;Do[s=s+1/k;le=Length[ContinuedFraction[s]];
> If[le==n,bb=Append[bb,k}];Break[]],{k,400}],{n,50}];bb
>  %O A107921 1
>  %K A107921 ,nonn,
>  %A A107921 Zak Seidov (zakseidov(AT)yahoo.com)
>   May 28 2005
> 
> 
> --- David Wilson <davidwwilson at comcast.net> wrote:
> >
> > ----- Original Message -----
> > From: "Leroy Quet" <qq-quet at mindspring.com>
> > To: <seqfan at ext.jussieu.fr>
> > Sent: Friday, May 27, 2005 3:12 PM
> > Subject: Continued Fractions of Harmonic Numbers
> >
> >
> > >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 the fundamental sequence in the OEIS, namely,
> > a(n) = number of terms in
> > continued fraction of nth harmonic number?  If so,
> > then your sequence can be
> > described as the least inverse of a.
> >
> > > 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 am sure that as n grows, the above-described a(n)
> > grows erratically
> > without bound (although I would be surprised if you
> > could come up with a
> > decent lower bound).  In its erratic hopping around,
> > it is not unthinkable
> > that a(n) misses some small value m, then eventually
> > n grows large enough
> > that a(n) > m, so that a(n) = m has no solution.
> > This is fairly common
> > behavior erratic sequences.
> >
> > For example, consider the sequence
> >
> > a(n) = SumOfDigits(3^n)/9
> >
> > for n >= 2.  This sequence grows with approximately
> > linear but erratic
> > growth.  a(n) takes on every value m < 62, but
> > apparently misses a(n) = m =
> > 62.  a(n) continues to grow until it becomes
> > apparent (but provable at best
> > with great difficulty) that a(n) will never again be
> > as small as 62.  So we
> > can confidently say that a^-1(62) does not exist.
> >
> > > 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
> >
> >
> 
> __________________________________________________
> Do You Yahoo!?
> Tired of spam?  Yahoo! Mail has the best spam protection around
> http://mail.yahoo.com
>