Yes. Your construction is better than mine. For N=5, 120=2x3x5, 119=7x, and 121=11x. So from 114 to 126 it is 2,5,2,3,2,7,2,11,2,3,2,5,2. A gap of 13. I only got a gap of 11 from 2312 to 2322 2,3,2,5,2,7,2,3,2,11,2.
For N=6 we have 210=2x3x5x7. 209=11x. But 211 isn't a multiple of 13...