So yes, while it's perfectly fine to run your pool pump without a filter cartridge installed, it's better to have a backup filter cartridge to ensure that you don't have to run it without one.
In most cases, you can operate the cartridge pool filter without the filter cartridge inside the tank just to maintain circulation as you wait for the cartridge to dry. However, doing so requires you to ensure you've removed all the internal parts and reassembled the tank tightly.
Short answer--yes. The first response, however, would be why would you use a pool without a filter? The sand filter or other filtration system is essential to the health and safety of the swimmers. It is also necessary to have a circulation pump for maximum chlorination.
When you cannot circulate water, the water stagnates and chemicals tend to settle out in the lower levels of the pool. It makes it almost impossible to add any chemicals because you don't have a good way to mix the water.
If you are without power, grab your pool brush! Not only will scrubbing the sides and bottom reduce or prevent algae, it also causes the water to move and circulate, helping maintain an even level of chlorine throughout the pool.
A pool can only be safe for swimming without a pump for a few days and a maximum of one week. The definite number of days the pool will be safe will also depend on the temperatures of the water, the weather at that particular time, the level of chlorine, and how clean the pool was before the pump stopped functioning.
To keep the pool clean without a filter, it is necessary to use chlorine with a flocculant or to use a flocculant chemical. This product groups the impurities that float in the water, causing them to fall to the bottom of the pool so that they can be removed later with a cleaner.
A swimming pool pump is a standard feature in the operation of swimming pools, that allow you to pump water through a filter, circulate the water and keep the water clean. A swimming pool pump is an essential component in ensuring you have clean and safe water to swim and bathe in.
I think the answer to your question is about 3-6 days. The problem is that the chlorine that you need to keep the bacteria in check is used up more quickly as the temperature rises, the activity increases, and as sweat and other body stuff is put into the pool.
Filter Size
The filter flow rate must be rated at least the same GPM as your pump, or higher. With pool filters, it's best to err on the larger size so it can handle the power of your pump. A helpful rule of thumb is to choose a filter with at least 1 square foot per 10,000 gallons pool capacity.