New Sequence from wrong Comment in A083088

[David W. Cantrell]

Rainer Rosenthal wrote:

> David W. Cantrell wrote:
>
>> For your sequence giving the greatest number of consecutive integer
>> reciprocals, beginning at 1/n, which may be added without exceeding 
>> 1,
>> I conjecture that
>>
>> a(n) = floor( (e - 1)n - (e + 1)/2 + (e + 1/e)/(12(2n - 1)) ) + 1
>                                                               ^^^^^^
>            ******* if I am correct, this "+1" is necessary ----/
>            *******
>            ******* we need a(2) = 2 because 1/2+1/3 < 1 < 
> 1/2+1/3+1/4
>            *******

Dear Rainer,

Many thanks for correcting my silly mistake!

And I am now quite embarrassed to say that I see a typographical error 
that I had made also: Instead of (e + 1/e), I had intended (e - 1/e).

With those two corrections, and using your correction to express the 
formula more nicely in terms of (n - 1/2), we should have

a(n) = floor( (e - 1)(n - 1/2) + (e - 1/e)/(24(n - 1/2)) )

> Dear David,
>
> numerical checks up to 3^12 verify your formula!

Please do at least a small "sanity check" for my doubly corrected 
formula. I sincerely apologize for my _two_ errors!

> The very fast computation of b(n) allowed another comparison to that
> "near-miss" sequence A083088 and I found:
>
>       a(n) - A083088(n-1)
>       -------------------   ---------> 
> 0.111750472724977109594431...
>              n               n -> oo

I assume that you actually intended to type

0.0111750472724977109594431...

which is easily deduced to be  e - 2 - 1/sqrt(2) .

> The constant is not in Plouffe's Inverter.

Indeed, a simple lookup fails to find it.

> The existence of this limit
> astonishes me because of the simple definition
>
>       A083088(n) = floor(n*(1+sqrt(2))/((1+sqrt(2))-1)) + 1;
>
> Am I right that this has something to do with Paul D. Hanna's 
> findings?
> So finally the wrong comment could be turned into an interesting 
> one,
> as there is a deeper relationship between A083088(n) and a(n) than 
> just
> the match between the first 24 values.

Alas, there is nothing deep about that constant. It just follows from 
the asymptotics of the two sequences. That's why I said "easily 
deduced" above.

Best regards,
David