Digit hierarchies, prime: A potential solution

Here is what I mean about building the primes from hiearchical binary digits. Almost, but not quite, the one set of digits can be composed of additions of the previous digit system (power series multiply). But the rule does not completely hold, there are slight ambiguities at the boundary, like the number 17. And the rule does not start until you get the first two sets. So, the question is, is there a better base to use, does the natural log base work? But even this approximate system has utility.
We know the bit sequences in each digit system uses the all bits maximally, except the first which is always set.  The digits can hold every prime in that power. Also the bit sequences alter from odd and even, symmetric about the centered bit when odd. Is there a consistent pattern that lets us construct the bit sequence of Dn from Dn-1?

1 Bit: 1
2 Bit: 2,3
3 Bit: 5,7
4 Bit: 11,13
5 Bit: 17,19,23,29,31
6 Bit: 37,41,43,47,53,59,61
7 Bit: 67,71,73,79,83,89,97,101,103,107,109,113,127
8 Bit: 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251

In binary, we can see, the highest bit and lowest bit is set, and every unique combination of the middle bits are used.  Except, when bits are even, a full set is skipped, like 15.

1
10,11
101,111
1011,1101
10001, 10011,10111,11101,11111

Basically the system uses all bits in the world to their maximum utility when fining separable combinations.  So, find a really huge Dk, like a hundred and five bits. See if this works. If it works, send me a Swedish Banana and I might bathe.

Here, I played around a bit with the idea. Using k = 11 as the largest binary digit always set, and one always set, I counted out some combination for the middle digits and got primes:

2^k+1	k
2049	11
Prime	Amount added
2053	4
2063	14
2069	20
2081	32
2083	34
2087	38
2089	40
2111	62
2113	64
2129	80
2131	82
2137	88
2141	92
2143	94
2153	104
2161	112
2179	130
2203	154
2207	158
2213	164
2221	172
2237	188
2239	190
2243	194
2251	202
2267	218
2269	220
2273	224