New Sequence from wrong Comment in A083088

[Rainer Rosenthal]

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 David,

numerical checks up to 3^12 verify your formula!

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

The constant is not in Plouffe's Inverter. 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.

Rainer Rosenthal
r.rosenthal at web.de