This is Sloane's A121559, which essentially iterates A064722. The behavior is controlled by the (appropriately weighted) distribution of prime gaps below N.
Heuristically, with $\ell=1-1/\log N$, you'd expect something like $(1-\ell)\left(1+\ell^3+\ell^5+\ell^7+\ell^{11}+\cdots\right)$ 1s below N, where the exponents are 1 less than the locations of the 1s. You can stop the sum around $\log^2 N$.