Erdos Woods Numbers -- #A059756

[hv at crypt.org]

victor at idaccr.org (Victor S. Miller) wrote:
:I have a new method of computing Erdos Woods Numbers (sequence
:A059756), and now have over 100,000 terms.  Clearly a sequence this
:big can't fit on OEIS.  Is there a good place to put them, so they can
:be acessible.  How long a sequence can the OEIS accomadate?

The OEIS description says:
  Erdos-Woods numbers: the length of an interval of consecutive integers
  with property that every element has a factor in common with one of the
  end-points.
[...]
  Example: a(1) = 16 refers to the interval 2184, 2185, ..., 2200.

I tried to find out a bit about these numbers; several web pages quoted
exactly the same description as OEIS with no additional information,
and the only other reference I found was an abstract in:
  http://www.mat.unisi.it/docenti/sorbi/home/pub/papers/RUSHEAD.pdf
.. which seems to have quite a different definition:

  Two sequences of positive integers (a, a+1 .. a+k) and (b, b+1 .. b+k)
  are called [an] 'Erdos-Woods pair' if for each i: 0 <= i <= k the integers
  a+i and b+i have the same prime factors; in this case we say also that
  the pair have a 'depth' k.
  The following is known as the 'Erdos-Woods Conjecture': for some positive
  integer k there is no Erdos-Woods pair of depth k.

Now 'a+i and b+i have the same prime factors' seems a much stronger
condition than 'a+i has a factor in common with one of the endpoints'.
Are these two definitions intentionally referring to two completely
different types of sequence? Or could one of them be mistaken?

Also, if the OEIS definition is correct, is it missing some additional
constraints? I don't understand, for example, why '1' is not in the
list.

Hugo