Erdos Woods Numbers -- #A059756

[Rainer Rosenthal]

 > I think the description in OEIS could do with
 > improvement.

I don't think so, because it's correct.

But I did learn a lot from looking closer at
what is written there.

When I did refer to B28 in UPINT, I falsy thought
the definition there to be equivalent to the one
for A059756. Looking closer at the comment

**R. K. Guy: Unsolved Problems in Number Theory, 1981,
           related to Sections B27, B28, B29.

we only have "related to". And indeed, there is no
mentioning of "endpoints" but only gcd(x,p) /= 1
for any x in the interval an p = product of the rest.

I will present my little table again, because there
was an error for number 2189:


      2   5   11  17  23  37  59  71  137   199   439   733    1097
        3   7   13  19  31  43  61  73   157   313   547   1093
--------------------------------------------------------------------------------------
2183                     . 1   1   .
2184  3 1   1    1       . .   .   .
2185      1          1 1 . .   .   .
2186  1                  . .   .   .                          1
2187    7                . .   .   .
2188  2                  . .   .   .                   1
2189           1         . .   .   .          1
2190  1 1 1              . .   .   . 1
2191        1            . .   .   .             1
2192  4                  . .   .   .    1
2193    1          1     . . 1 .   .
2194  1                  . .   .   .                              1
2195      1              . .   .   .                1
2196  2 2                . .   . 1 .
2197             3       . .   .   .
2198  1     1            . .   .   .       1
2199    1                . .   .   .                      1
2200  3   2    1         . .   .   . 1
2201                     1 .   .   1

Any two numbers x, y in 2184 ... 2200 have a linking number z in this
interval, i.e. gcd(x,z) /= 1 and gcd(y,z) /= 1.
The "endpoint"-condition is fulfilled too, which is nice to see.

 > If I've (now) correctly understood, we need only have a common
 > factor with _one_ endpoint. so for the interval [2, 3], 2 has a
 > common factor with 2, and 3 with 3.

Oh yes, sorry, you are right! The OEIS says that a(1) = 16
is chosen because we have interval k, k+1, ... k+16  (k=2184) such
that each of these numbers has a factor in common with one of the
endpoints k or k+16.

Consequently one should allow a(0) = 1, because the interval k, k+1
(with e.g. k=2) does have the same property: each number has a factor
in common with one of the endpoints k or k+1.

Hmmm, *something* will have to be changed indeed, I think.

Best regards,
Rainer Rosenthal
r.rosenthal at web.de