I suspect the following statement is true and I can use it in my work if it is true. However I am not a number theorist and I could not prove it myself. I was wondering if this is known to number theorists.

Statement: For each positive real number $\alpha$ there exist a natural number $N$ such that, for every $n \geq N$ each of the intervals $[n^{\alpha},2n^{\alpha}) , [2n^{\alpha},3n^{\alpha}), ...$ contains at least one natural number that has no common factor with $n$.