7 14 31 51 70 81 104 107 129 149 155 175 192 215 238 259 279
299 301 301 314 328 350 372
Inspired by the discussion with the author of, and the clues to its solution are in that discussion: What is the last integer in this sequence?
I have constructed a finite (terminating), non decreasing sequence (I purposefully allowed this and not strictly increasing) that has a unique solution.
Not only is the solution unique, but it has a strong personal meaning for me.
I first intended to give the solution here immediately, as I very much doubt that it will be found, part of the challenge is to find the reason it is the unique solution.
However before posting it, I decided to let it run for a while as an open question first.