sensor - Dust visible on pictures with small aperture

A few months ago I took my then new camera (Nikon D7200) out with a tripod to take some small aperture shots. I noticed a significant amount of artifacts on the resulting pictures. A quick Google search informed me that this was sensor dust and to be expected at these apertures. My camera only had 1600 shutter actuations when I first noticed the spots and both the lenses I used were new. Below are two examples of pictures with the spots. The spots are easiest to see against the sky, but you may have to zoom in to see them. The spots are nonexistent when using apertures larger than about f/18. Is this normal for a DSLR, even when new? Is there a way to prevent/minimize these artifacts at small apertures without post-processing?

First image: f/36, 1 second, 140mm (first lens)

Second image: f/22, 30 second, 32mm (second lens + variable ND filter)

Answer

Is there a way to prevent/minimize these artifacts at small apertures without post-processing?

Clean the sensor. Or more specifically, clean the front of the filter stack directly in front of the sensor. If you're not comfortable doing it yourself, have it professionally cleaned.

Is this normal for a DSLR, even when new?

For many recent Nikon models it has been. Nikon tends to be sure everything inside the camera's light box that needs lubrication leaves the factory with a healthy dose of lubrication. Sometimes the excess gets slung off by mechanical movement and can wind up on the sensor. If the camera is still under warranty Nikon may perform a cleaning at no charge. It all depends on what geographic world area you live in and the policies of Nikon for your service region.

For all interchangeable lens cameras sensor dust is an issue that must be managed. Even if the camera is immaculately clean when fresh from the factory, over time dust will find its way onto the sensor. Cleaning the sensor can be scary for someone who has never done it before, but it is fairly easy to learn and doing it yourself can save a lot in terms of shipping/insurance costs as well as fees for paying someone else to clean it.