This starts to look like prime k-tuplets; in our case 9-tuplets.
I started reading on the concept of admissible 9-tuplets, and why some patterns are not admissible.
This is the fun of pushing what at first appears to be a silly puzzle ;-)
I made the problem up myself as I was teaching a small group how to solve the simpler problem of placing the numbers 1 to 9 in the same triangle. That puzzle has a neat insight that cuts the solutions down enormously.