New Sequence from wrong Comment in A083088

[Rainer Rosenthal]

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