On covering sets of a special type by geometric progressions with a certain restrictions

The paper investigates the classic problem of covering the start of the natural number series with the minimum number of geometric progressions under various constraints (on the starting point, progression step, and non-intersection of progressions). Among similar problems, the following should be noted: covering arithmetic progressions with geometric progressions with real-valued steps, covering the start of the natural number series with geometric progressions with a fixed number of terms and a real-valued step, and covering the start of the natural number series with geometric progressions with a rational step. Thus, the uniqueness of the work lies in the constraints imposed on geometric progressions, particularly that the step is a natural number. Optimal solutions were found for cases where: the step constraint is 2, the step constraint is 2 with a prohibition on intersection, and the starting point constraint is 1. Lower bounds were obtained for cases where: there are no constraints, there is a prohibition on intersection, and there is a step constraint of 3. Upper bounds were obtained for cases where: there are no constraints, and there is a prohibition on intersection.