Does this sequence have any mathematical significance?

Solution 1:

If I use abc to generate the sequence it comes out like this:

abc                       3
bc                        5
c bc                      8
c c bc                    12
bc c c bc                 18
c bc bc c c bc            27
c c c bc c bc bc c c bc   41

The list of numbers on the right, the sequence length up to that point, is A061419. A Mills-type formula for this sequence has the constant $\frac{2}{3}K(3)$, where $K(3)$ is A083286, and this last constant is related to the Josephus problem.

So your sequence is related to prime number sieves and the prime number theorem, which are closely linked with the Josephus problem. That you could not find the sequence in the OEIS is most likely because mathematics in this area lags behind other fields of number theory.