Erdős–Woods number

Consider a sequence of consecutive positive integers $[a,\; a+1,\; dots\; a+k]$. The length "k" is an Erdős-Woods number if there exists such a sequence in which each of the elements has a common factor with one of the endpoints, i.e. if there exists a positive integer "a" such that for each integer "i", $0\; le\; i\; le\; k$, either $gcd(a,\; a+i)\; >\; 1$ or $gcd(a+i,\; a+k)\; >\; 1$.

The Erdős-Woods numbers are listed as OEIS|id=A059756. The first few are given as:16, 22, 34, 36, 46, 56, 64, 66, 70though arguably 0 and 1 could also be included as trivial entries. OEIS2C|id=A059757 lists the starting point of the corresponding sequences.

Investigation of such numbers stemmed from a prior conjecture by Paul Erdős:

:There exists a positive integer "k" such that every integer "a" is uniquely determined by the list of prime divisors of $a,\; a+1,\; dots\; a+k$.

Alan R. Woods investigated this for his 1981 thesis, and conjectured that whenever "k > 1", the interval $[a,\; a+k]$ always included a number coprime to both endpoints. It was only later that he found the first counterexample, $[2184,\; 2185,\; dots\; 2200]$ with $k\; =\; 16$.

David L. Dowe proved that there are infinitely many Erdős-Woods numbers, and Cégielski, Heroult and Richard showed that the set is recursive.

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