The most famous example of 16 as an Erdos-Woods number is the range of 16 consecutive integers starting with 2184.

Another $n$ for $k = 16$ is 2044224, which we obtained by multiplying 2184 by 936. The factorization is $2044224 = 2^6 \times 3^3 \times 7 \times 13^2$ while $2044224 + 16 = 2044240 = 2^4 \times 5 \times 11 \times 23 \times 101$ The table of factorizations

shows that each of the numbers in this range shares at least one factor with one if not both of the numbers capping the range.

Next we have a slightly longer example, this one for $k = 34$ The smallest matching $n$ is 47563752566, a squarefree number with a factorization of $2 \times 11 \times 17 \times 23 \times 41 \times 157 \times 859$ The number capping the end of the range is the decidedly non-squarefree 47563752600, with a factorization of $2^3 \times 3^2 \times 5^2 \times 7 \times 13 \times 17 \times 19 \times 29 \times 31$ While the size of these numbers forbids verification on your typical pocket calculator, these numbers are well within the reach of a Javascript implementation of trial division. Here we could be tempted to omit the even numbers, as they obviously share 2 as a prime factor with the range start and the range end, as well as multiples of 3 or 5 as they thus share factors with the range end. But, on the hope that it turns out to be at least a little bit instructive, the factorizations of all the numbers in our chosen range is given.