ErdősWoods number
An integer $k$ is an ErdősWoods number if there is an integer $n$ such that each of the consecutive integers $n+i$ for $$ shares at least one prime factor^{} with either $n$ or $n+k$. In other words, if for a $k$ there is an $n$ such that each evaluation of $\mathrm{gcd}(n,n+i)>1$ or $\mathrm{gcd}(n+k,n+i)>1$ returns true, then $k$ is an ErdősWoods number.
For example, one $n$ for $k=16$ is 2184. 2184 is ${2}^{3}\times 3\times 7\times 13$, while $2184+16=2200={2}^{3}\times {5}^{2}\times 11$. We then verify that
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2186 is even and so shares 2 as a factor with both 2184 and 2200.

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2187 is 3 more than 2184 and therefore must also be divisible by 3. In fact, it is ${3}^{7}$.

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2188 is even and so shares 2 as a factor with both 2184 and 2200, suggesting we needn’t look at any other even numbers^{} in this range.

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2189 is 11 less than 2200 and therefore must be divisible by 11. In base 10 we can quickly verify that 2 + 8 = 10 and 1 + 9 is also 10.

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2191 is 7 more than 2184 and thus must be divisible by 7.

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2193 is 9 more than 2184 and thus divisible by 3.

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2195 is obviously divisible by 5.

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2197 is 13 more than 2184 and thus must be divisible by 13. In fact, it is ${13}^{3}$.

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2199 is 15 more than 2184 and thus divisible by 3.
Knowing one $n$ for a given $k$ one can find other $n$ by multiplying the odd prime factors of $n+k$ (just once each, let’s call that product “$q$”) and then calculating $n(jq+1)$ with $j$ any positive integer of one’s choice. To give one example: with $n=2184$ and $j=17$, we get another $n$, namely 2044224. The range 2044224 to 2044240 displays the same patterns of factorization as described above, except that 2044227 and 2044237 are a semiprime and a sphenic number^{} respectively, as opposed to 2187 and 2197 which are both prime powers.
Other ErdősWoods numbers are 22, 34, 36, 46, 56, 64, 66, 70, 76, 78, 86, 88, 92, 94, 96, 100, etc., listed in A059756 of Sloane’s OEIS (the smallest odd ErdősWoods number is 903), while A059757 lists the smallest matching $n$ for each of those $k$.
References
 1 R. K. Guy, Unsolved Problems in Number Theory^{} New York: SpringerVerlag 2004: B28
Title  ErdősWoods number 

Canonical name  ErdHosWoodsNumber 
Date of creation  20130322 17:37:14 
Last modified on  20130322 17:37:14 
Owner  PrimeFan (13766) 
Last modified by  PrimeFan (13766) 
Numerical id  4 
Author  PrimeFan (13766) 
Entry type  Definition 
Classification  msc 11A05 
Synonym  ErdosWoods number 
Synonym  ErdösWoods number 