The ordinary Jacobsthal function $j$ is defined by setting $j(n)$ as the smallest number $m$, such that for each consecutive $m$ numbers in the integers, there is at least one of the numbers comprime to $n$. There are estimes for $j$, for example $$j(n) \ll \log ^2 (n)$$ and it is conjectured by Jacobsthal, that we could improve this to $$j(n) \ll ( \log (n) / \log ( \log (n)) )^2.$$ See for example http://www.tcnj.edu/~hagedorn/papers/JacobPaper.pdf .

We define now $h(n)$ as the smallest number $m$, such that for each square $a \in \mathbb{N}$, the sequence $$a+1, a+1, \ldots, a+m$$ contains at least one number comprime to $n$. My question is now, are there any good upper estimates for $h(n)$ ? Clearly, $h(n) \leq j(n)$ holds, so we could take the above estimate, but I am searching for something stronger, perhaps like $$h(n) \leq C \log(n),$$ where $C=2$ or so.

The ordinary Jacobsthal function $j$ is defined by setting $j(n)$ as the smallest number $m$ such that, for each consecutive $m$ integers, at least one of the numbers is coprime to $n$. There are estimates for $j$; for example, $$j(n) \ll \log ^2 (n)$$ and it is conjectured by Jacobsthal that we could improve this to $$j(n) \ll ( \log (n) / \log ( \log (n)) )^2.$$ See for example http://www.tcnj.edu/~hagedorn/papers/JacobPaper.pdf .

We define now $h(n)$ as the smallest number $m$ such that, for each square $a \in \mathbb{N}$, the sequence $$a+1, a+1, \ldots, a+m$$ contains at least one number comprime to $n$. My question is now, are there any good upper estimates for $h(n)$ ? Clearly, $h(n) \leq j(n)$ holds, so we could take the above estimate for $j(n)$, but I am searching for something stronger, perhaps like $$h(n) \leq C \log(n),$$ where $C=2$ or so.

The ordinary Jacobsthal function $j$ is defined by setting $j(n)$ as the smallest number $m$, such that for each consecutive $m$ numbers in the integers, there is at least one of the numbers comprime to $n$. There are estimes for $j$, for example $$j(n) << \log ^2 (n)$$ and it is conjectured by Jacobsthal, that we could improve this to $$j(n) << ( \log (n) / \log ( \log (n)) )^2.$$ See for example http://www.tcnj.edu/~hagedorn/papers/JacobPaper.pdf .

We define now $h(n)$ as the smallest number $m$, such that for each square $a \in \mathbb{N}$, the sequence $$a+1, a+1, \ldots, a+m$$ contains at least one number comprime to $n$. My question is now, are there any good upper estimates for $h(n)$ ? Clearly, $h(n) \leq j(n)$ holds, so we could take the above estimate, but I am searching for something stronger, perhaps like $$h(n) \leq C \log(n),$$ where $C=2$ or so.

The ordinary Jacobsthal function $j$ is defined by setting $j(n)$ as the smallest number $m$, such that for each consecutive $m$ numbers in the integers, there is at least one of the numbers comprime to $n$. There are estimes for $j$, for example $$j(n) \ll \log ^2 (n)$$ and it is conjectured by Jacobsthal, that we could improve this to $$j(n) \ll ( \log (n) / \log ( \log (n)) )^2.$$ See for example http://www.tcnj.edu/~hagedorn/papers/JacobPaper.pdf .

We define now $h(n)$ as the smallest number $m$, such that for each square $a \in \mathbb{N}$, the sequence $$a+1, a+1, \ldots, a+m$$ contains at least one number comprime to $n$. My question is now, are there any good upper estimates for $h(n)$ ? Clearly, $h(n) \leq j(n)$ holds, so we could take the above estimate, but I am searching for something stronger, perhaps like $$h(n) \leq C \log(n),$$ where $C=2$ or so.