Recurrence relation for A043569?

[Alonso Del Arte]

I've just sent through the form a Mathematica command to calculate the
terms of A043569. It involves multiplying Mersenne numbers by powers
of 2 and sorting. It works but it's not terribly elegant.

Is there a recurrence relation for this sequence? I've tried several
different things along the lines of

a(1) = 2, a(n) = a(n -1) + log_2 a(n - 1) + log_2 a(n - 2) etc., etc.,

and they work except after a(n - 1) becomes a power of two, or before.
Any suggestions?

Al