Erdos Woods Numbers -- #A059756

[Rainer Rosenthal]

hv at crypt.org wrote:

>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; ... webpages ...
>
You may have a look into R. K. Guy's UPINT, third edition,
page 127:  http://makeashorterlink.com/?I1CA52BBB
(on-line lookup in amazon.com - a real GREAT feature!)

This is chapter B28 and can be found on page 83 in the
second edition. The problem is stated as follows:

     This is related to the problem: find n consecutive
     integers, each having a composite common factor
     with the product of the other n-1. If the composite
     condition is relaxed, and one asks merely for a
     common factor greater than 1, then 2184+i (1<=i<=17)
     is a famous example.

>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.
>
Because neighbours never have a common factor > 1.
(But thanks for your question. I was lost at first, too.)

It was a fun project for me to construct the following table
for the said interval between 2183 and 2201:

In the following table you find the prime factorization, e.g.
you see  2193 = 3^1 * 17^1 * 43^1  or  2192 = 2^4 * 137^1.
Please note that 2183 and 2201 share no factor with any of
the numbers in the interval. On the other hand you will find
a partner (i.e. one with a common factor) for each of the
numbers in the interval. An extreme candidate is 2197 = 13^3,
which is supported only by 2184 = 13 * something.

_____________________________________________________________________

      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
_____________________________________________________________________

Table 1: Illustration for Sloane's Sequence A059756

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