Constant C=0.1688... for A081881 seems to be wrong

[David W. Cantrell]

Rainer Rosenthal wrote:
> David W. Cantrell wrote:
>
>> You seem to have forgotten what I wrote on Jan. 9 in your sci.math
>> thread "Inverse Harmonic Series". I not only confirmed the few
>> digits that BC had given for the constant, but I also gave many
>> more. Furthermore, I gave a conjectured formula for the sequence.
>
> You are right and I am sorry for the mistakes I made.
> In this post I want to ask one more question about BC's constant
> and I also want to tell about a strange response from Superseeker.
>
>> Of course, since we can now calculate more elements of the sequence
>> very easily, we can also specify C with much greater accuracy.
>> Indeed, it seems that we can even give a formula for the sequence
>>
>> a(n) = 1 + floor(C*exp(n))                      (*)
>>
>> where
>>
>> C = 0.1688563566671442037316797755009010341...
>
> I tried in vain to understand how you arrived at (*). I could have
> asked in sci.math but I think it appropriate to ask here instead
> of hopping elsewhere.

There's not much to explain. In
<http://www.research.att.com/~njas/sequences/A081881>,
Benoit Cloitre had already said that "a(n) is asymptotic to C*exp(n)
where C=0.1688...", which I suppose that has been proven. My simple
numerical investigation seemed to indicate that
0 < a(n) - C*exp(n) < 1.
I don't know if that's always true, but if it is, then of course (*)
holds for all n.

Best regards,
David