Continued Fractions of Harmonic Numbers

[zak seidov]

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 
> 
> 

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