Given the positive integers $ n $ and $ k, consider the finite sequence of consecutive integers $ n, n + 1, dotsc, n + k-1 $, noted by the interval $[n,n+k-1]$. We would like to find a maximum subset $ {a_1, a_2, dotsc, a_t } $ it's in relatively prime pair. Set the maximum size (not only maximum) per $ F (n, k) = max t $. We are interested in the minimum value of $ F (n, k) $ more than $ n $, that is to say., $ s (k) = min_n F (n, k) $. In other words, we are interested to know that among all the $ k positive integers, how many coprime numbers per pair we can make sure to find.

In the document entitled "Completing the First Integral Integer Subsets" of P. Erdős and J. L. Selfridge on the Proceedings of the Manitoba Conference on Digital Mathematics, Winnipeg (1971), it was found that $ s (k) geq k ^ { frac {1} {2} – epsilon} $. However, they also mention that the true value of $ s (k) $ is probably closer to $ frac {k} { log k} $. Our question is: is there a way to improve this situation? $ k ^ { frac {1} {2}} $? Is there any other literature that has worked on this problem?

Another related question. In the same sequence over consecutive integers $[n,n+k-1]$we would like to find a minimal set of prime numbers $ {p_1, p_2, dotsc, p_ tau } $ which "covers" the entire interval, that is to say each integer of $[n,n+k-1]$ is divisible by some $ p_i $, or $ 1 leq i leq tau $. Set minimum size (not only minimal) $ g (n, k) = min tau $. We are interested in the minimum value of $ g (n, k) $ more than $ n $, that is to say., $ t (k) = min_n g (n, k) $. In other words, we are interested to know that among all the $ k positive integers, how many prime numbers we must have to cover the interval.

It's obvious that $ t (k) geqs (k) $. Again, our question is this: Is it true that $ t (k) geq k ^ { frac {1} {2}} $?