New Sequence from wrong Comment in A083088

[David W. Cantrell]

Dear SeqFan and especially dear Rainer,

I have a conjectured formula for your new sequence and also a report
of another wrong comment by the author of the previously mentioned
wrong comment.

-----------------------------------------

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)) )

Perhaps one could give a heuristic argument (similar to that mentioned
in http://www.research.att.com/~njas/sequences/A002387) that, with
high probability, my formula should always work.

-----------------------------------------

The wrong comment mentioned previously here was submitted by
Robin Saunders. On the same day that comment was submitted,
he also submitted a comment about A054414, stating that

a(n) = A083088(n) + n - 1

First, he surely intended to submit a(n) = A083088(n) + n instead.
That error was probably causing by thinking that an offset was 1, when
it was actually 0.

But even with that correction, A054414(n) = A083088(n) + n is
incorrect, first failing for n = 24.

Thus, his comment for A054414 should be removed.

-----------------------------------------

Best wishes to all for the new year!
David


----- Original Message ----- 
From: "Rainer Rosenthal" <r.rosenthal at web.de>
To: <Seqfan at ext.jussieu.fr>
Cc: "Paul D. Hanna" <pauldhanna at juno.com>
Sent: Friday, January 04, 2008 23:43
Subject: New Sequence from wrong Comment in A083088


> Dear SeqFan and especially dear Paul
>
> We know that 1/2+1/3 < 1 < 1/2+1/3+1/4.
> I wondered how many terms would form a maximal
> sum 1/n + 1/(n+1) + ... not exceeding 1.
> Calling this number of terms a(n), the above
> example shows a(2) = 2.
>
> The first 24 elements of this sequence are
> identical to those of sequence A083088. But
> the 25-th element and others don't fit. The
> comment there is wrong (not Paul's fault!).
>
> To correct this, I would like to submit the correct
> sequence
> -------
> 1, 2, 4, 6, 7, 9, 11, 12, 14, 16, 18, 19, 21, 23,
> 24, 26, 28, 30, 31, 33, 35, 36, 38, 40, 42, 43,
> 45, 47, 48, 50, 52, 54, 55, 57, 59, 61, 62, 64,
> 66, 67, 69, 71, 73, 74, 76, 78, 79, 81, 83, 85
>
> together with the Maple program
> -------
> Harmonic := n -> expand(Psi(n+1))+gamma;
> a := proc(n) local erg,m; erg := 0; for m to
> infinity do if Harmonic(n-1+m)-Harmonic(n-1) > 1
> then erg := m-1; break; fi; od; return erg; end;
> seq(a(n),n=1..50);
>
> For the example section I am planning to write:
> -------
> a(3)=4 because the reciprocals of 3,4,5,6 sum to
> 19/20 < 1 and those of 3,4,5,6,7 sum to 153/140 > 1.
>
> In the comment section it might be nice to have a
> reference to A083088 because of the many equal elements:
> -------
> Values a(1) up to a(24) are identical to sequence A083088,
> but a(25)=42.
>
> Please have a look at this and let me know if I
> made something wrong. I would like to make a
> correct submission on the first shot.
>
> As soon as my new sequence will show up in the OEIS
> it would be nice to cancel the wrong comment in A083088.
> This would be best done by Paul himself, I think.
>
> One last point: A097682 uses sqrt(2) in the Digamma-context
> and I am a bit uneasy about this after the "near miss" in
> A083088. Is A097682 waterproof?
>
> Happy New Year to all SeqFans and good health to Neil
> and everybody,
>
> Rainer Rosenthal
> r.rosenthal at web.de
>